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m
n
A.M ERI C A.1S[
Journal of Mathematics
PURE AND APPLIED.
Editor in Chief,
J. J. SYLVESTER, LL. D., F. R. S., Gorr, Mem. Inst, of France.
Associate Editor in Charge,
WILLIAM E. STORY, Ph. D., [Leipnc.)
With the Co-operation of
BENJAMIN PEIRCE, LL. D., F. R. S., SIMON NEWCOMB, LL. D., F. R. S.,
Pkofkssor of Matiikmatics in Uakvard Cobr. Mem. Inst, of France,
University, Superintendent of the American Ephemeris,
In jIVEeclianics,
In Astronomy,
AND
H. A. ROWLAND, C. E.,
In I^hysics.
Published under the Auspices of the
JOHNS HOPKINS UNIVERSITY.
Ylavta ya [.tav ra yiyvocfxofisva dpi^fidv ej^ovri. — Philolaoa.
"Volume I.
BALTIMORE:
Printed for the Editors by John Murphy & Co.
B. Westermann & Co., \ i^ y k '^' Williams & Co., Boston.
D. Van Nostrand, J * Fbrree & Co., Philadelphia.
Trubner & Co., London, A. Asher & Co., Berlin,
Gauthier-Villars, Paris,
1878.
, ^ « •
f
f ♦
\V
424976
• •
• > «
• ••
«
t •
«
• • •
INDEX TO VOLUME I.
Cay LEY, A. — Desiderata and Suggestions. paok
No. 1. The Theory of Groups, 60
No. 2. The Theory of Groups ; Graphical Representation, . . . .174
A Link-Work for a:*. (Extract from a Letter to Mr. Sylvester), .... tJ86
Clifford, W. K. — Remarks on the Chemico-Algebraical Theory. (Extract from a
I^'tter to Mr. Sylvester), 126
Applications of Grassmann's Extensive Algebra, 350
Craio, T. — The Motion of a Point upon the Surface of an Ellipsoid, .... 359
Dixon, T. 8. E. — A New Solution of Biquadratic Equations, . . . . .283
Eddy, H. T.— The Theorem of Three Moments, 27
The Elastic Arch, 241
The Two General Methods in Graphical Statics, . . . . . . . 322
Engler, E. a. (and Holman, M. L.) — The Tangent to the Parabola, . . . . 379
Frankland, a. — Remarks on Chemico-graphs. (Extract from a Letter to Mr.
Sylvester), 345
Franklin, F. — Bipunctual Coordinates, 148
On a Problem of Isomerism, . . . , . . . . • . 365
Note on Indeterminate Exponential Forms, 368
Glashan, J. C. — An Extension of Taylor's Theorem, 287
Halsted, G. B. — Bibliography of Hyper-Space and Non-Euclidean Geometry, . .261
Addenda to the preceding Paper, 384
Hammond, J. — On the Mechanical Description of the Cartesian, 283
Hill, G. W. — Researches in the Lunar Theory. Chapters I and II, . . 5, 129, 245
Holman, M. L. (and Engler, E. A.) — The Tangent to the Parabola, .... 379
Kendall, O. H. — On a Short Process for Solving the Irreducible Case of Cardan's
Method, 285
LiPSCUiTZ, R. — Demonstration of a Fundamental Theorem obtained by Mr. Sylvester, 336
Loudon, J. — Euler's Equations of Motion, 387
Condition of a Straight Line Touching a Surface, 388
Lucas, E. — Tli6orie des Fonctions Num6riques Simplement Pcriodiqucs, . . 184, 289
Mallet, J. W. — Some Remarks on a Passage in Professor Sylvester's Paper as to the
Atomic Theory, 277
MuiR, J. — Letter to Mr. Sylvester on the word Continuant, 344
Newcomb, S. — Note on a Class of Transformations, which Surfaces may undergo in
Space of more than Three Dimensions, 1
Peiuce, C. S. — Exposizione del Metodo dei Minimi Quadrat!. Per Annibale Fer-
BERO, Tenente (hlonnello di Stato ilaggiore, cc, Firenze, 187G, . . .59
• « •
111
IV
INDEX TO VOLUME I.
Page
Phillips, A. W. — A Link-Work for the Lemniscate. (Extract from a Letter to Mr.
Sylvester), ............. 386
Rowland, H. A. — Note on the Theory of Electric Absorption, 53
Story, W. E.— On the Elastic Potential of Crystals, 177
Sylvester, J. J. — On an Application of the New Atomic Theory to the Graphical
Representation of the Invariants and Covariants of Binary Quantics^ . . 64
Appendix 1, On Diiferentiants Expressed in Terms of the Differences of the Roots
of their Parent Quantics, . 83
Appendix 2, Note on M. Hermite's Law of Reciprocity, . . . ; .90
Note -4. Completion of the Theory of Principal Forms, 106
Note B, Additional Illustrations of the Law of Reciprocity, .... 107
Note C. On the Principal Forms of the General Sextinvariant to a Quartic and
Quartin variant to a Sextic, • • - 112
Note -D. On the Probable Relation of the Skew Invariants of Binary Quintics and
Sextics to one another and to the Skew Invariant of the same Weight of the
Binary Nonic, 114
Appendix 3, On Clebsch's " li^infachstes System associirter Formen" and its
Generalization, * 118
Note on the Ladenburg Carbon-graph, 125
Note on the Theorem Contained in Professor Lipschitz's Paper, (see pp. 336-340), . 341
A Synoptical Table of the Irreducible Invariants and Covariants to a Binary
Quintic, with a Scholium on a Theorem in Conditional Hyperdeterminants, 370
Weichold, G. — Solution of the Irreducible Case, 32
. Historical Data Concerning the Discovery of the Law of Valence, . . 282
ERRATA.— </n addition to those given on page 125.)
Pju?e 20, lines 10 and 12, dele period after valves of ^ and p| .
Page 21, line 6, insert period at end of line.
I'aKe 26, line 2, dele periotl.
I*agc 33, in the last right-hand member of the collective equation,
for p' read py
Page 36, Vlll./or p\p'l read p'p'J .
Page 30, X, for - — read
P"P' 1
«'2«' I
P P l\
Ps^e 37, in the lastitymbol of the denominator ol the fourth right-,
hand member of the symbolical equation,
Pa^ 41, line 14,/or — 179 + 4V^=nt read — 179 f- 42» I~3.
Page 42, line :i,for 346 read 846.
Page 42, line 7, for \ read — J.
Page 44, first line. /or ppi read p'p^ ^
Pa-e 44, lines 3 and A,fo.- — 133l'8U21 read - 13398SII21.
Page 41, line 3 from bottom,/w p^^[t\ read pp\ .
i»age 4.\ lines 2, 3, \for — 357K —3 read —55711' ^=¥.
Pago 46, first rme,/or B V p^ + Pj read B f pj {- p\ .
Page 4S, firat line of the value of c,for p—p read p — p'.
Page 53, line 9, for vol. 2 read vol. I, part 2.
Page 89, line 2 of J\)stseript,for March 14 read Feb. 14.
Page 17S, line 9 from bottom, /or elf, *Z read <H , cZ .
Page 178, line 4 " " for JI + reati U f .
Page 178, line 3 " " for Z -f read Z -f .
Page 183, line 3 " " forU, H, Z, readU,U , Z.
Page 246, line 22, for (5) read (—5).
Page 249, lino 3, /or aj iro// aj.
Page 252, last Mne, for -^ read — .
Page 255, line 4 from bottom, add second arm of parenthesis at the
end of the line.
Page 256, line 2,/or - ^ 'o~ ^ 'I <U • • •) '^«<' - ~ -^ ~ ' v^ ( r -J .
Page 256, line 4 from bottom,/or jrrr ,^-rzx read — ^ -
*'' 19958400 ll>9r>84<X)
Page 257, line 10 from bottom, /or 2 • 34902 read 2 • 43902 .
Page 869, line 11,/ormehrer read mehrerer.
Page 273, line 22,/or Frkyk read Frkuk.
Page 274, lino 12, /or Litber, read Lit. Ber.
Page 275, line 18, /or Monroe reatl Monro.
Page 282, line 2,/or 1875 read 1878.
Page 336, in the name of the author, /or B. read R.
).
NOTICE TO THE READER.
In presenting to the Public this first nunfiber of the American Journal
OF Mathematics, the editors think it advisable, in order to prevent disap-
pointment on the part of subscribers and contributors, to state briefly the
principles by which they will be guided in its management.
Although in the first instance designed to supply a want, as a medium
of communication between American mathematicians, its pages will always
be open to contributions from abroad, and promises of support from various
•l
foreign mathmeticians of eminence have already been received.
The publication of original investigations is the primary object of the
Journal. In addition to this, from time to time concise abstracts will be
inserted of subjects to which special interest may attach, or which have been
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Critical and bibliographical notices and reviews of the most important
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been valuable aids to them in scientific investigation. This difficulty the
editors hope to some extent to remove.
The Journal will be published in volumes of about 384 quarto pages,
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but for the present separated as nearly as may be found practicable by
intervals of three months. In order that the numbers may not diffier too
much in size, an article will be made, when necessary, to run on from one
number to the next, as is the practice with foreign journals of a similar
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• . *
111
•
iv Notice to the Reader,
This is the only journal of the kind in the English language published
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mathematical papers are too well understood to need enumeration.
The editors believe it will materially aid in fostering the study of
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due to the enlightened liberality of the Trustees of the Johns Hopkins
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It is to be understood that there will be no problem department in
the Journal, but important remarks, however brief, may be inserted as
notes. Persons desirous of offering to the Public mathematical problems
for solution, are recommended to send them to
''The Analyst^' edited and published by J. E. Hendricks, I)es Moines,
loxoa; or to
*' The Mathematical Visitor^'' edited and published by Artemas Martin,
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The subscription price of the Journal is $5.00 a volume, and single
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Communications and subscriptions (by postal money order) may be
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((
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Brunswick, "
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NOTE ON A CLASS OF TRANSFORMATIONS WHICH SURFACES
MAY UNDERGO IN SPACE OF MORE THAN
THREE DIMENSIONS.
By Simon Newcomb.
If the material bodies which surround us were placed in a space of more
than three dimensions, their kinematic susceptibilities would be increased in
a manner which, at first sight, would seem very extraordinary. Each body
would, in fact, be susceptible of n independent forward motions, and ^ ~ ^
separate rotations, n being the number of dimensions of the space. My
present purpose is not, however, to discuss the general theory of the subject,
but to point out a special case of it as seen in a remarkable transformation to
which closed surfaces may be subjected in space of four dimensions. The
proposition in question may be expressed as follows :
If a fourth dimension were added to space^ a closed material surface {or
shell) could be turned inside out by simple ^exure ; without either stretching or
tearing.
For simplicity we may suppose the surface to be spherical. Let
X, y, z, u,
be the general rectangular coordinates in the supposed space of four
dimensions. An infinite plane space of three dimensions may then be repre-
sented by the equation
ax -\- by -\- cz -\' du •=. A,
a, ft, etc., being any constants whatever. For simplicity we may suppose a, b
and c all equal to zero, and the axes of x^ y and z therefore to lie in the space
of three dimensions under consideration. An Euclidian or natural space
may then be represented by the single equation uz= A^ A being an arbitrary
constant. The four-dimensional space may be divided into an infinity of
Euclidian spaces by giving aH possible values to A.
* Vol. I— No. 1. 1
• ••
* •
• •
k •.
• •
•^«
•.-\
fc
•. •:•.•
•• /• 2 Newcomb, Note on a Class of Transformations.
To define a surface in four-dimensional space by rectangular coordinates,
two equations are necessary. If the surface is one which can exist in three-
dimensional space, one of these equations must be of the first degree in x^ y,
etc. For example, the most general equations of the sphere in four-dimen-
sional space are
{x-ay + {y-by + {z-cy + {u-dy = r^ ;
(1) ax + ^y + YZ + huz=k,
m
a, /3, y, 5 and k being constants, and a, 6, c, rf, the coordinates of the centre.
This centre is not necessarily situated in the same Euclidian space with the
surface ; in fact there is a series of points, each of which is equidistant from
every point of the surface, but only one of them lies in the same Euclidian
space with the surface. This point is. one whose coordinates fulfil the con-
dition
aa + /3ft + yc + 5rf = ^,
or,
a{x — a) +^{y — b) +y{z — c) +h{u — d) =0.
If we choose our axes of coordinates so that the equations shall have the
simplest forms, putting
a = /3 = y = 0, 5 = 1,
the general equations (1) will become
{x-ay + {y-by +\z-cy + {u-dy ^7^',
u =z k.
Now, to consider the transformation of a- material spherical surface, we
must consider this surface as an indefinitely thin shell, situated between two
surfaces. We shall suppose the natural space in which the sphere is situated
in the beginning to be conditioned by the equation
t* = ^ = 0,
and we shall take the centre of the sphere as the origin of coordinates. The
equations of the inner surface of the spherical shell, which we may call -4,
will then be of the form
and of the outer one, B
iXn + yi + A = r{\ u = 0.
Now suppose that, the outer surface remaining fixed, we move the inner one
in the direction of the axis of te by a small quantity k^ allowing its radius
Newcomb, Note on a Class of Transformations. 3
at the same time to vary in such a way that the thickness of the shell shall
remain unchanged. Its equations may then be expressed in the form
s' + f + z' + u''= {r + 8ry ; u = k,
and we must, if possible, determine 8r so that the thickness of the shell shall
remain unaltered. To find the new thickness, we remark that the square of
the distance of any point of the outer from any point of the inner surface is
(2) {s-x^y + {y-yrY + {z-z,y + uK
The condition that this shall be a minimum for a given point of the outer
surface is
(x — x^) dx + {y — yx) dy + {z — z^) dz = 0,
dx^ dy and dz being subject to the condition
xdx + ydy + zdz = 0.
The simultaneous existence of these equations depends upon our having
X y z
Xi~ yi~ zi'
or,
(3) X = Xj;i , y=z^,, z = 7iZi,
X being a quantity which differs very little from unity. These values of x^ y
and z, being substituted in (2), give for the square of the thickness of the
shell after the motion, of the inner surface
{^-ly i^i+yi+zi) + i^ = {x-iyri + if' .
Let us put A = Ti — r, for the original thickness of the shell. In order that
the thickness may remain unaltered, it is necessary and sufficient that we
determine it.and X simultaneously in terms of an arbitrary angle d by the
conditions
(4) k = hsin d'j 1 — ^ = — cos 0.
The original position of the inner surface will be that corresponding to 6 = 0.
Suppose, now, that 6 increases from 0° to 180°, k and X being constantly
determined by the condition (4). It is then evident that the shell will expe-
rience no other deformation than that arising from flexure, the flexure
involving a stretching of the outer surface which may be made indefinitely
small by diminishing the thickness of the shell. When 6 reaches 180° we
shall once more have ^ = 0, so that the surface A will be brought back into
Newcomb, Note on a Class of Transformations.
its original natural or Euclidian space. Moreover, ^ being then greater than
unity, the equations (3) show that the radius of this surface A^ or or^ + y^ + z^,
will be greater than that of B.
The outer surface cci'\-yi + z\=zri will therefore be the inner one, and
the other the outer one, the change being brought about by flexure alone.
By the motion we have supposed, every point of the surface A has described
a semicircle round the corresponding point of B^ 6 representing the angle of
position of the line joining the two points during the motion. For simplicity
we have supposed the surface A only to vary, the flexure taking place round
Bj but we might equally have supposed a mean surface to remain constant.
RESEARCHES IN THE LUNAR THEORY,*
m
By G. W. Hill, lifyack Turnpike, N. Y.
When we consider how we may best contribute to the advancement of
this much-treated subject, we cannot fail to notice that the great majority of
writers on it have had before them, as their ultimate aim, the construction
of Tables: that is they have viewed the problem from the stand-point of prac-
tical astronomy rather than of mathematics. It is on this account that we
find such a restricted choice of variables to express the position of the moon,
and of parameters, in terms of which to express the coefficients of the peri-
odic terms. Again, their object compelling them to go over the whole field,
they have neglected to notice many minor points of great interest to the
mathematician, simply because the knowledge of them was unnecessary for
the formation of Tables. But the developments having now been carried
extremely far, without completely satisfying all desires, one is led to ask
whether such modifications cannot be made in the processes of integration,
and such coordinates and parameters adopted, that a much nearer approach
may be had to the law of the series, and, at the same time, their convergence
augmented.
Now, as to choice of coordinates, it is known that, in the elliptic theory,
the rectangular coordinates of a planet, relative to the central body, the axes
being parallel to the axes of the ellipse described, can be developed, in terms
of the time, in the following series,
arzza 2 -T «^<« cos ?y,
<== — 00 % "2"
<= + co Y {i-V)
y = b^ -- /^ sin t^,
<=»— CO I ■«■
a and h being the semi-axes of the ellipse, e the eccentricity, g the mean
anomaly and, for positive values of f, the Besselian function (in Hansen's
notation)
A- 1.2... t L l.(i + l) "^ 1.2.(i+l)(i + 2) "'J'
* Communicated to tho National Academy of Sciences at the session of April, 1877.
6 Hill, Researches in tlie Lunar Theory.
while, for negative values of f, we have
(-1) (0
i A — A '
and, for the special ease of i =z 0, we put the indeterminate
e.
T •^o — T
Here the law of the series is manifest, and the approximation can easily be
carried as far as we wish. But the longitude and latitude, variables em-
ployed by nearly all the lunar- theorists, are far from having such simple
expressions ; in fact, their coefficients cannot be expressed finitely in terms of
Besselian functions. And if this is true in the elliptic theory, how much
more likely is a similar thing to be true when the complexity of the problem
is increased by the consideration of disturbing forces? We are then justified
in thinking that the coefficients of the periodic terms in the development of
rectangular or quasi-rectangular coordinates are less complex functions of
their parameters than those of polar coordinates. There is also another
advantage in employing coordinates of the former kind ; the differential
equations are expressed in purely algebraic forms ; while, with the latter,
circular functions immediately present themselves. For these reasons I have
not hesitated to substitute rectangular for polar coordinates.
Again, as to parameters, all those who have given literal developments,
Laplace setting the example, have used the parameter m, the ratio of the
sidereal month to the sidereal year. But a slight examination, even, of the
results obtained, ought to convince any one that this is a most unfortunate
selection in regard to convergence. Yet nothing seems to render this para-
meter, at all desirable, indeed, the ratio of the synodic month to the sidereal
year would appear to be more naturally suggested as a parameter. Some
instances of slow convergence with the parameter m may be given from
Delaunay's Lunar Theory: the development of the principal part of the
coefficient of the evection in longitude begins with the term -^ me, and ends
•i.!- A.i_ J. 413277465931033 o • • .■* • • i a. ^ xv ••/¥» • a.
With the term i5i88>38080 ^^' ^g^^"? ^^ the principal part of the coefficient
of the inequality whose argument is the difference of the mean anomalies
21
*of the sun and moon, we find, at the beginning, the term -j- me(f, and, at the
end, the term — 35389^4 — vi^eff. It is probable that, by the adoption of some
function of m as a parameter in place of this quantity, whose numerical
Hill, Researches in the Lunar Theory. 7
value, in the case of our moon, should not greatly exceed that of m, the
f(U*egoing large numerical coefficients might be very much diminished. And
nothing compels us to use the same parameter throughout ; one may be used
in one class of inequalities, another in another, as may prove most advan-
tageous. It is known what rapid convergence has been obtained in the series
giving the values of logarithms, circular and elliptic functions, by simply
adopting new parameters. Similar transformations, with like effects, are,
perhaps, possible in the coefficients of the lunar inequalities. However, as
far as my experience goes, no useful results are obtained by experimenting
with the present known developments ; in every case it seems the proper
parameter must be deduced from a priori considerations furnished in the
course of the integration.
With regard to the form of the differential equations to be employed,
although Delaunay's method is very elegant, and, perhaps, as short as any,
when one wishes to go over the whole ground of the lunar theory, it is subject
to some disadvantages when the attention is restricted to a certain class of
lunar inequalites. Thus, do we wish to get only the inequalities whose coeffi-
cients depend solely on 7n, we are yet compelled to develop the disturbing
function R to all powers of e. Again, the method of integrating by unde-
termined coefficients is most likely to give us the nearest approach to the
law of the series ; and, in this method, it is as easy to integrate a differential
equation of the second order as one of the first, while the labor is increased
by augmenting the number of variables and equations. But Delaunay's
method doubles the number of variables in order that the differential
equations may be all of the first order. Hence, in this disquisition, I have
preferred to use the equations expressed in terms of the coordinates, rather
than those in terms of the elements ; and, in general, always to diminish the
number of unknown quantities and equations by augmenting the order of the
latter. In this way the labor of making a preliminary development of R in
terms of the elliptic elements is avoided.
In the present memoir I propose, dividing the periodic developments of
the lunar coordinates into classes of terms, after the manner of Euler in his
last Lunar Theory,* to treat the following five classes of inequalities : —
1. Those which depend only on the ratio of the mean motions of the
sun and moon.
**Th^ria Moiuum Lunce^ nova meihodo pertractata. Petropoli^ 1772,
8 Hill, JResearches in the Lunar Theory.
2. Those which are proportional to the lunar eccentricity.
3. Those which are proportional to the sine of the lunar inclination.
4. Those which are proportional to the solar eccentricity.
5. Those which are proportional to the solar parallax.
A general method will also be given by w*hich these investigations may be
extended so as to cover the whole ground of the lunar theory. My methods
have the advantage, which is not possessed by Delaunay's, that they adapt
themselves equally to a special numerical computation of the coefficients,
or to a general literal development. The application of both procedures
will be given in each of the just mentioned five classes of inequalities, so
that it will be possible to compare them.
I regret that, on account of the difficulty of the subject and the length
of the investigations it seems to require, I have been obliged to pass over
the important questions of the limits between which the series are con-
vergent, and of the determination of superior limits to the errors committed
in stopping short at definite points. There cannot be a reasonable doubt
that, in all cases, where we are compelled to employ infinite series in the
solution of a problem, analysis is capable of being perfected to the point of
showing us within what limits our solution is legitimate, and also of giving
us a limit which its error cannot surpass. When the coordinates are devel-
oped in ascending powers of the time, or in ascending powers of a parameter
attached as a multiplier to the disturbing forces, certain investigations of
Cauchy afford us the means of replying to these questions. But when, for
powers of the time, are substituted circular functions of it, and the coefficients
of these are expanded in powers and products of certain parameters pro-
duced from the combination of the masses with certain of the arbitrary
constants introduced by integration, it does not appear that anything in the
writings of Cauchy will help us to the conditions of convergence.
Chapter I.
Differential Equations. — Properties of motion derived from JacoMs integral.
We set aside the action of the planets and the influence of the figures
of the sun, earth and moon, together with the action of the last upon the
Hill, Researches in the Lunar Theory, 9
sun, as also the product of the solar disturbing force on the moon by the
small fraction obtained from dividing the mass of the earth by the mass of
the sun. These are the same restrictions as those which Delaunay ha«
imposed on his Lunar Theory contained in Vols. XXVIII and XXIX of the ^ ^^
Memoirs of the Paris Academy of Sciences. The motion of the sun, about
the earth, is then in accordance with the elliptic theory, and the ecliptic is
a fixed plane.
Let us take a system of rectangular axes, having its origin at the centre
of gravity of the earth, the axis of x being constantly directed toward the
centre of the sun, the axis of y toward a point in the ecliptic 90° in advance
of the sun, and the axis of z perpendicular to the ecliptic. In a'ddition, we
adopt the following notation : —
f =: the distance of the sun from the earth ;
W =. the sun's longitude ;
ft zz the sum of the masses of the earth and moon, measured by the
velocity these masses produce by their action, in a unit of time,
and at the unit of distance;
m' =. the mass of the sun, measured in the same way ;
n' =, the mean angular velocity of the sun about the earth ;
a! zr the sun's mean distance from the earth.
In accordance with one of the above-mentioned restrictions we have the
equation : —
The axes of x and y having a velocity of rotation in their plane, equal
to — , it is evident that the square of the velocity of the moon, relative to the
at
earth's centre, has for expression, in terms of the adopted coordinates,
= W -^^Tt —dt — + 'W (^+^)'
The potential function, in terms of the same coordinates, is
_ |M n^ a!^ n'' af^
~ V(^ + 3^ + /) + V[(r'-^)M^7HMl ^^ ^''•
If the second radical in this expression is expanded in a series proceeding
according to descending powers of r', and the first term omitted, since it
20 JIjU- JU-M^^nh^.* *h T** Lrt-.kiK" 7'kf*{r%
>>■ ^^ «- ^i • •
»• If «•» ■ • • • • - I
-. J" — ■:».- r - r- - i r - r- - ;
r- - ^ " - J
.r ij" r - r - - s r - r - ]
Slii-.:*^ iiitr iifftr^tiitl *•: ul:ijji:* c nj.cj.a trt A lit f 'Til
6
fj
f- ri_2
CT
i a ^•3
^ fifii -cIlz. ii •=iiit!?esBt»L. «ivi :*f "UH: TtrlLiue? wrSrl ieiire lift p:»sh5 <i: -.d'
ir^
CT €.T
f-r- • iLJ
•
^^
iG.
6-*
liZ
Kr»r ':iJU*n#*nitHin -tf id*: muut iitnTii^i. tJI lit lirr:.^, :i. iLe 1b>: en 'resa-Mi c«f
Hill, Researches in the Luiuir Theory. 11
When all the inequalities, involving the solar eccentricity, are neglected,
the equations admit an integral in finite terms. For, in this case, we have
and H' does not explicitly contain t ; hence, multiplying the equations sever-
ally by the factors dx, dy^ and dz^ and adding the products, both members
of the resulting equation are exact differentials. Integrating this equation,
we have
dx'+ du" + dz^
~Tii zz il' + a constant.
This integral equation appears to have been first obtained by Jacobi.*
As it will be frequently referred to in what follows, I shall take the liberty
of calling it Jacobi's integral.
If we take two imaginary variables
n has the following simple expression, being a function of two variables only,
s/ns V ('^ —u)s/{f — s) 2f'
If this is expanded in descending powers of r', and, as before, the term
omitted,
VMS r V. J
+ ^^, [am« + Am*s + t\«^ + A«=']
n a! r n
+ ^2 7B [i " "* + " "'* + »'' ^^'^ + " "* + "» * J
+ • •
The additional variable, necessary to complete the definition of the moon's
position, may be so taken that the expression of T may be simplified as much
as possible. This expression may be written
^ rp_duds _ . { ydz—zdnY . ^d^ xdy—ydx , <?^'^„„_ ,2\
" dt' ^\u—sYdt' '^ dt dt ^ df^' ^'
* Comptea Rendus de V Academic des Sciences de Paris. Tom, iii, p. 69.
12 Hill, Researches in the Ldinar Theory.
There does not seem to be any function of x^ y and z^ which, adopted as a
new variable to accompany u and «, would reduce this to a very simple form.
However, when we are engaged in determining the inequalities independent
of the inclination of the lunar orbit, this transformation will be useful to us.
For, in this case, js = 0, and the values of u and s become
u =1 X -\- y s/ — 1,
s = X — y V — 1,
and T is given by the equation
o rr duds dX uds — sdu . dX^
2 1 = — -^ ; + — — us.
df dt dt ^ df
Although n is expressed most simply by the systems of coordinates we
halve just employed, the integration of the differential equations will be
easier, if we suppose that the axes of x and y have a constant instead of a
variable velocity of rotation, the axis of x being made to pass through the
mean position of the sun instead of the true. To obtain the expression for
T correspondent to this supposition, it is necessary only to write w' for — in
the former values. As for H, it can be written thus
where
r^=ar^ + y^ + z^ = the square of the moon's radius vector ;
S = X cos i; + y sin i; ;
i; z= the solar equation of the centre.
This function Jbeing expanded in a series of descending powers of r', as before,
we have
+ . .
Hill, Beaearches in the Lunar Theory. 13
And the corresponding differential equations are
^'f _ 2/i' ^ — ^-'
df dt dx '
^+2n'^ = ^,
de ^ dt dy '
(Pz _d£r
df ~ dz '
Thus much in reference to the equations under as general a form as we
shall have occasion for in the present disquisition. We shall now suppose
that they are reduced to as restricted a form as is possible without their
becoming the equations of the elliptic theory ; that is, we shall assume that
the solar parallax and eccentricity and the lunar inclination vanish. With
these simplifications, in the first system of coordinates,
rp, _ (/,r* + df , xdy—ydx
and, in the second,
rp, duds n' uds — sdu
~ 2df 2 di '
£1' = -t= + !«'* {u + sy .
And the differential equations, correspondent, are, in the first case,
-d¥ ^''dt^\:y ^"^ r-^'
and, in the second,
The Jacobian integral has severally the expressions
Mr r
14 Hill, JResearcIies in the Lunar Theory.
The terms — 2n' -J^. 2ri — , &c., have been introduced into the equations
at at
by making the axes of coordinates movable ; but since the putting of n' =
makes the solar disturbing force vanish, there is no inconsistency in attribu-
ting them to the solar action. Then, in the case of the vanishing of this
action, we have the Equations of ordinary elliptic motion
m
Thus, in the restricted case we consider, all the terms, added to the diflfer-
ential equations of motion by the solar action, are linear in form and have
constant coeflBcients. This, and the circumstance that t does not explicitly
appear in the equations, are two advantages which are due to the particular
selection of the variables x and y. If -^ were constant, the equations would
be linear with constant coefficients and easily integrable.
The constants ft and w' can be made to disappear from the differential
equations, if, instead of leaving the units of length and time arbitrary, we
assume them so that (i = lj and n' = 1. The new unit of length, will then
be equal to f^^ units of the previous measurement. The equations, thus
simplified, are .
df r^
It will be perceived that, in this way, we make the differential equations
absolutely the same for all cases of the satellite problem.
dx
Let us put p = 3", then
dt
dt
with their integral
Hill, Researclies in the Lunar Theory. • lo#
Or, by making y the independent variable,
dx p
^y [| + 3x^-26/-p^]*
^ = 2
LM^
The probleni is then reduced to the integration of two diflTerential equations
of the first order. Were this accomplished, and p eliminated from the two
integral equations, we should have the equation of the orbit. If we put
the differential equations can be written in the canonical form,
dx__ dW
dy dp
dp _ dW
dy dx
It may be worth while to notice also the single partial differential
equation, to the integration of which our problem can be reduced. Returning,
to the arbitrary linear and temporal units, and, for convenience, reversing
the sign of C, if a function of x and y can be found satisfying the partial
differential equation
and involving a single arbitrary constant h, distinct from that which can be
joined to it by addition, the intermediate integrals of the problem will be
dx dV , , dy dV
- - = -:r— + n y, -^ z= — nXj
dt dx '^' dt dy '
and the final integrals
dV dV ^ ,
a and c being two additional arbitrary constants. The truth of this will be
evident if we differentiate the four integral equations with respect to t and
compare severally the results with the partial differential coefficients of the
partial differential equation with respect to ar, y, h and C.
16 Hill, Bemearch's in the Liniar Th'ort/.
Although, in this manner, the problem seems reduced to its briefest
terms, yet, when we essay to solve it, setting out with this partial differential
equation, we are led to more complex expressions than would be expected. It
would be advisable, in this method of proceeding, to substitute polar for
rectangular coordinates, or to put
x = r cos <p, y=-^ »in ^.
The partial differential equation, thus transformed, is
T^' + [7 %- «''•]' = T + ^ ""'' + -^'+ ' ""'' '^' ^*'
This would have to be integrated by successive approximations, and it is
found that this method, which, at first sight, seems likely to afford a briefer
solution of the problem, because but one unknown quantity was to be deter-
mined, and this free from the variable /, and involving only half of the number
of arbitrary constants introduced by integration, when developed, leads to as
complex operations as the older methods, and, moreover, has the disadvantage
of giving results which need prolix transformations before the coordinates can
be exhibited in terms of the time.
Although we shall make no use of the equations in terms of polar
coordinates, they may be given here, for the sake of some special proi>erties
they possess in this form. They are
r^',^%- 2n'r % + ^- 3«' V cos^ = 0,
ill fit' fit r
^K^+"')]+>""''™-'* =«•
with their integral
rfr r
By dividing the second of the differential equations by r*, the variables are
separated, and X denoting the longitude of the moon, we have
^dt
jK' being a constant. Thus, after the longitude is determined in terms of <,
the radius vector is obtained by a quadrature. But it can also be found,
without the necessity of an integration, by dividing the integral by r* and
IIiLL, Researches in tJie Lunar Theory, 17
%•
1 </r^
then eliminating the term — - -~, by means of its value derived from the
r^ dv
second differenticil equation ; in this way we get
^ _C _
r^ v
f ^^'1 + f «'^ sin 2^
+ ^%-rn'-'-co^'^.
As we desire to make constant numerical application of the general
theory, established in what follows, to the particular case of the moon, we
dehiy here, for a moment, to obtain the numerical values of the three
constants /^, n and C. The value of ^i may be derived either from the
observed value of the constant of lunar parallax combined with the mean
angular motion of the moon, or from the intensity of gravity at the earth's
surface and the ratio of the moon's mass to that of the earth. We will adopt
the latter procedure. The value of gravity at the equator, g = 9.779741
metres, the unit of time being the mean solar second. We propose, however,
to take the mean solar day as the unit of time, and the equatorial radius of
the earth as the linear unit. This number must then be multiplied by
6377sy7T5' (6377397.15 metres is Bessel's value of the equatorial radius.)
Moreover, the theory of the earth's figure shows that, in order to obtain the
attractive force of the earth's mass, considered as concentrated at its centre
of gravity, a second multiplication must be made by the factor 1.001818356.
With our units then this force is represented by the number 11468.338: and
the moon's mass beinf; taken at q, ,.,»-, of the earth's, her attractive force is
o 81.022/7 '
represented by the number 140.676. Consequently
It = 11609.0M.
The sidereal mean motion of the sun in a Julian year is 1295977".4r516,
whence
n = 017202124.
The value of C might be obtained from the observed values of the moon's
coordinates and their rates of variation at any time. However, as the eccen-
tricity of the earth's orbit is not zero, C obtained in this manner would be
found to undergo slight variations. The mean of all the values. obtained
in a long series of observations might be adopted as the proper value of this
18 II ILL, Researches in the Lunar Theory.
quantity when regarded as constant. But it* is much easier to derive it
approximately from the series
2C = {unY^ [1 + 2m — Mn^ — nv» — ^^^ m' — ^ m' — ?5 Jl m^' — \lll m"] ,
which will be established in the following chapter. Here n denotes the
moon's sidereal mean motion, and m is put for ,. In this formula the
n — n
terms which involve the squares of the lunar eccentricity and inclination and
of the solar parallax are neglected ; this, however, is not of great moment,
as, being multipled by at least m'-, they are of the fourth order with respect
to smallness. The observations give n = 0.22997085, hence
C= 111.18883.
If it is proposed to assume the units of time and length so that ti and
n may both be unity, it will be found that the first is equal to 58.13236
mean solar days, and the second to 339.7898 equatorial radii of the earth.
The corresponding value of C is 3.254440.
Let us now notice some of the properties of motion wiiich can be derived
from Jacobi's integral. This integral gives the square of the velocity relative
to the moving axes of coordinates; and, as this square is necessarily positive,
the putting it equal to zero gives the equation of the surface which separates
those portions of space, in w^hich the velocity is real, from those in which
it is iifiaginary. This equation is, in its most general form,
V (•I*" + ^ + z") s^l{a: — xy+i/- + Z-] 2
which is seen to be of the 16tli degree. ' As y and z enter it only in even
powers, the surface is symmetrically situated with respect to the planes of xy
and xz. The left. member is necessarily positive, (the folds of the surface, for
which either or both the radicals receive negative values, are excluded from
consideration), hence the surface is inclosed within the cylinder whose axis
passes through the centre of the sun perpendicularly to the ecliptic, and
whose trace on this plane is a circle of the radius
«'V(3 + -?^-).
As, in general, the second term of the quantity, under the radical sign, is
much smaller than the first, this radius is, quite approximately V3 a , Thus,
1
in the case of our moon, assuming — ^ = sin 8" 848, we have this radius =
^ a
t
Hill, Researches in the Lunar Theory, 19
V3.001383a'. It is evident that, for all points without this cylinder, the
velocity is real ; and as, for large values of 2, whether positive or negative,
the left member of the equation becomes very small, it is plain that the
cylinder is asymptotic to the surface. Every right line, perpendicular to the
ecliptic, intersects the surface not more than twice, at equal distances from
this plane, once above and once below.
Let us now find the trace of the surface on the plane of xy. Putting
p for the distance of a point on this trace from the centre of the sun,
and it is evident that the cubic equation,
will give the limits between which the values of p can oscillate. If C is
negative, this equation has but one real root which is negative; consequently,
in this case, the surface has no intersection with the plane of xy. But, in all
the satellite systems we know, C is positive, and this condition is probably
necessary to insure stability. Hence we shall restrict our attention to the
case where C is positive. Then all the roots of the equation are real, and
two are positive. It is between the latter roots that p must always be found.
To compute them, we derive the auxiliary angle Q from the formula
L n^a J
or, since differs but little from 90°, with more readiness from
^ "T^"?^' 1 + 3 + ^j
n^a'i. n^a n^a*J
cos^O =
[1 + S -#7.1 '
C
or, as —T-To is a small quantit^^ with sufficient approximation from
n ^a ^
cos 6 =
^^.
c
l + §
c
The two roots are then
c \ . /^^„ e
p, = 2af^ (1 + § -y sin (60° - |) .
20
Hill, Hesearclies in tlie Lunar Theory.
The trace of the surface on the plane of ay is then wholly comprised in the
annular space between the two circles described from the centre of the sun as
centre with the radii' pi and pg. Moreover, as in most satellite systems we
have -f ,3 equal to a very small fraction, (for our moon ^-^—^ = -__~ V
it is plain that, for points whose distance from the earth is comparable with
their distance from the sun, the trace is approximately coincident with these
circles. For the term -^, in the equation, may then be neglected in com-
r
parison with the other terms.
In the case of our moon there is found
e = 87° 62' 11".53.
and hence
pi = 22815.15. p2 = 23816.09,
and, if r and p are regarded as the variables defining the position of a point
in the plane ay, the following table gives some corresponding values of these
quantities, for each of the tw^o branches of the trace approximating severally
to the two circles.
r.
433.325t
228T8 G9
r.
439.T922
P-
23761.81
450
228T6.n
450
23753.37
500
22869.68
500
23760.04
600
22860.13
600
23769.85
1000
22841.59
1000
23788.87
10000
228n.T0
10000
23813.43
46127.T0
22815.68
47127.55
23815.53
The first and last values correspond to the four points where the curves
intersect the axis of .r on the hither and thither side of the sun. It will be
seen that the approximation of the branches to the circles is quite close,
except in the vicinity of the earth, where there is a slight protruding aw^ay
from them.
In addition to these two branches of the trace, there is, in the case where
C exceeds a certain limit, a third closed one about the origin much smaller
than the former. As the coordinates of points in this branch are small
fractions of a', its equation may bo written, quite approximately.
r
I n'V.
-t-
Hill, Researches in the Lunar Theory. 21
It intersects the axis of ^ at a distance from the origin very nearly
II
and the axis of x at points whose coordinates are^the smallest (without regard
to sign) roots of the equations
X a! — X
_ ^^.^LJ^ = ^+ 3 ^'2^2_ J ,^,2 /^_^N2
X aC — X
For the mooit these quantities have the values
yo = 104.408, x^ = — 109.655, x^ = +109.694.
This branch then does not differ much from a circle having its centre at the
origin, more closely it approximates to the ellipse whose major axis = X2 — Xi,
and minor axis = 2yo-
The value of the coordinate 2, for the single intersection of the surface
with the axis of z above the plane of ory, is given by the single positive root
of the equation
if + -^y = C+ n^'afK
For the moon the numerical value of this root is
Zo = 102.956.
The intersection of the surface with the perpendicular to the plane of xy
passing through the centre of the sun is, in like manner, given by the equation
u ^J^^c+in''a:\
having but a single positive root, which is nearly
ia!
Zf,=
1 + ^Z.^
or, with less exactitude,
^0 = 3 «'. '
From these investigations it is possible to get a tolerably clear idea of
the form of this surface. When C exceeds a certain limit, it consists of three
separate folds. The first being quite small, relatively to the other two, is
closed, surrounds the earth and somewhat resembles an ellipsoid whose axes
have been given above. The second is also closed, but surrounds the sun,
6
22 IT ILL, lieaearrhes in the Lnimr Theory.
and has approximately the form of an ellipsoid of revolution, the seniiaxis in
the plane of the ecliptic being somewhat less than a', and the semiaxis of
revolution perpendicular to the ecliptic and i)n»sing through the sun being
about two-thirds of this. Tliis fold has a protuberance in the portion neigh-
boring the earth. The third fold is not closed, but is asymptotic to the
cylinder mentioned at the beginning of the investigation of the surface. Like
the second, it also is nearly of rev<dution about an axis passing through the
centre of the sun and perpendicular to the ecliptic. The radius of its trace
on the ecliptic is about as much greater than a\ as the radius of the trace
of the second fold falls short of that quantity. The fold. has a^rotuberance
in the portion neighboring the earth, and which projects towards this body.
The whole fold resembles a cylinder bent inwards in a zone neighboring the
ecliptic.
What modifications take place in these folds when the constants involved
in the equation of the surftice are made to vary, will be clearly seen from
the following exposition. Let us, for brevity, put
n -a -
and, for the moment, adopt a', the distance of the earth from the sun, as the
linear unit, and transfer the origin to the centre of the sun, and moreover put
n
Then the intersections of the surface, w^ith the axis of x^ will be given by the
two roots of the equation
. x^ — x' — hx'' + (* + 2 — 2y) .?• — 2 = 0,
which lie between the limits and 1 ; by the two roots of
x' — x" — hx" + (A +'-' + 2y) .r — 2 = 0,
which lie between 1 and sjh ; and by the two roots of
x' — X' — hjr + (/I _ 2 — 2y) x + 2 = 0,
which lie between and — j^h.
Hence, if C diminishes so much that the first of these three equations
has the two roots, lying between the mentioned limits, equal, the first fold
will have a contact with the second fold ; and if C fall below this limit, the
roots become imaginary, and the two folds become one. Again, if C is
diminished to the limit where the second equation has the mentioned pair of
roots equal, the first fold will have a contact with the third; and when 6' is
Hill, Researches in the Lunar Theory. 23
less than this, these two folds form but one. And when C is less than both
these limits, there will be but one fold to the surface.
In the spaces inclosed by the first and second folds the velocity, relative
to the moving axes of coordinates, is real ; but, in the space lying between
these folds and the third fold, it is imaginary ; without the third fold it is
again real. Thus, in those cases, where C and y have such values that the
three folds exist, if the body, whose motion is considered, is found at any
time within the first fold, it must forever remain within it, and its radius
vector will have a superior limit. If it be found within the second fold, the
same thing is true, but the radius vector will have an inferior as well as a
superior limit. And if it be found without the third fold, it must forever
remain without, and its radius vector will have an inferior limit.
• Applying this theory to our satellite, we see that it is actually within the
first fold, and consequently must always remain there, and its distance from
the earth can never exceed 109.694 equatorial radii. Thus, the eccentricity
of the earth's orbit being neglected, we have a rigorous demonstration of a
superior limit to the radius vector of the moon.
In the cases, where Cand y have such values that the surface forms but
one fold, Jacobi's integral does not afford any limits to the radius vector.
When we neglect the solar parallax and the lunar inclination, the pre-
ceding investigation is reduced to much simpler terms. The surface then
degenerates into a plane curve, whose equation, of the sixth degree, is
r
It is evidently symmetrical with respect to both axes of coordinates, and is
contained between the two right lines, whose equations are
^ - ± V M^z '
3/»'
and which are asymptotic to it. It intersects the axis of y, at two points,
whose coordinates are
The cubic equation,
*=±^
3 20 , 2/1 f.
r r 4- — — =
24
Hill, Researches in the Lunar Theory.
gives the values of r, for which the curves intersect the axis of x. If
{2C)% > %n',
these roots become equal. And if
(2C)i < V',
there are no real roots between these limits, and the curve has no intersection
with the axis of x. The figures below exhibit the three varieties of this curve.
FJ^.!
FI0.3
Fig, 1 represents the form of the curve in the case of our moon. In Fig. 2
we see that the small oval of Fig. 1 has enlarged and elongated itself so as to
touch the two infinite branches ; while, in Fig. 3, it has disappeared, the
portions of the curve, lying on either side of. the axis of x, having lifted
themselves away from it, and the angles having become rounded off. In Fig.
1, the velocity is real within the oval, and also without the infinite branches,
but it is imaginary in the portion of the plane lying between the oval and
these branches. Hence, if the body be found, at any time, within the oval,
it cannot escape thence,- and its radius vector will have a superior limit ; and,
if it be found in oniB of the spaces on the concave side of the infinite branches,
it cannot remove to the other, and its radius vector will have an inferior
limit.
Hill, Researches in the Lunar Theory. 26
In the case represented in Fig. 2, the same things are true, but it seenTs
as if the body might escape from the oval to the infinite spaces, or vice versa,
at the points where the curve intersects the axis of x. However, at these
points, the force, no less than the velocity, is reduced to zero. For the
distance of these points from the origin is the positive root of the equation
or
y 2^ _ f'dfin'
and this value is the same as that given by the equation
JL _ 3n'' = 0.
In consequence the forces vanish at these two points, and thus we have two
particular solutions of our diflferential equations *
In the case represented in Fig. 3, the integral does not aflPord any
superior or inferior limit to the radius vector.
The surface, or, in the more simple case, the plane curve, we have
discussed, is the locus of zero velocity ; and the surface or plane curve, upon
which the velocity has a definite value, is precisely of the same character and
has a similar equation. It is only necessary to suppose that the C of the
preceding formulae is augmented by half the square of the value attributed
to the velocity. Thus, in the case of our moon, it is plain the curves of equal
velocity will form a series of ovals surrounding the origin, and approaching
it, and becoming more nearly circular as the velocity increases.
Applying the simple formulae, where the solar parallax is neglected, to
the moon, we find that the distance of the asymptotic lines, from the origin, is'
^.
^^ = 500.4992.
The distance of the points on the axis of x, at which the moon would remain
stationary with respect to the sun, is
^JL = 235 5971.
^3ft'*
♦The corresponding solution, in the more general problem of three bodies, may be seen in the
Mieanique CiUste^ Tom, IV, p. 810.
7
. J".-:*^;- • *•" ./ 'if ,_.»«':' .*-!"(.
"l»t t i;,...( "• t IJ
'. -i; I". . 4-%>^Lr.33. . I'r^ ^L.:
THE THEOREM OF THREE MOMENTS.
By Hei^ry T. Eddy, 0. E., Ph. D., University of Cincinnati.
The theorem expressing the relationship between the moments of flexure
of a straight elastic girder at three successive points of support, was first
published by Clapeyron in the Comptes Rendus, 1857. The investigation
contemplated a girder of uniform cross section, whose points of support were
situated upon the same level at any unequal distances apart, and loaded with
any uniform loading between the successive points of support, but that
loading of different intensity in the different spans into which the points of
support divide the girder. The formula has since been generalized by the
labors of Winkler, Bresse. Heppel, Weyrauch and others so as to include all
cases ; viz : when the loading is distributed in any manner whatever, either
continuously or discontinuously ; when the cross section of the girder changes
uniformly or at intervals ; when the points of support are at any different
heights, and when the girder is fixed at the end supports in such a manner
that it can or cannot change its direction at one or both of those points. The
formula supposes that the elastic limits are in no case surpassed, either by too
great loading or too great difference in height between the points of support,
or both.
It is proposed in the present paper to obtain, in a direct manner, a new
and simplified equation to express this relationship in its most general form,
and to point out at the same time what may be regarded as the proper
significance of the quantities appearing in the equation.
Let us assume the ordinary fundamental equations applicable to elastic
girders under the action of vertical forces as already proven : they may be
stated as follows:
Let be any point of the girder at which we are to express either the
shearing stress, the bending moment, the curvature, the slope, or the deflection.
Let z = the distance from as an origin to the point of application of
any vertical force P.
Let x = the distance from as an origin to the point of application of
any bending moment M.
27
28 Eddy, The Theorem of Three Moments.
Let r = the radius of curvature of the deflected girder at 0.
Let / = the moment of inertia of the ctoss section of the girder about
its neutral axis at any point.
Let E = the modulus of elasticity of the material, supposed to be
constant.
Then the relations we propose to assume are :
Force applied at any point is P. ^
Shearing stress at is /S = 2 (P). I • • • • • • • (1)
Bending moment at is Jf = 2 {Pz). )
Curvature at is 1 -=- r = !ff = Jf -=- El. ^
SlopeatOis r=2(i2) = 2(if-T-^/). I (2)
Deflection at is i> = 2 {Ex) = 2 [Mx ^ EI). )
It is evident that P bears the same kind of relation to M in (1), that M-^ EI
does to i> in (2). In order to comprehend what the relationship is, it is only
necessary to conceive that vertical ordinates be laid off at each point of the
span equal to M\ the locus of their extremities has been called the "curve of
the actual moments," and it is known that it is the same as the catenary of
the same depth which supports the weights P. Now from this curve, another
curve (or polygon as the case may be) can be described, whose ordinates are
equal to R =z M -^ EI^ which may be called the "curve of the effective
moments," since the amount of bending (i. e. difference of slope) and the
deflection are dependent upon R. If this curve of effective moments be
regarded as the surface of some species of homogenous loading whose depth
is R, and a third curve be drawn such that its ordinates are the moments
which would be produced by such loading, then the third curve is the "curve
of actual deflections," and is the shape which the deflected girder will assume
under the action of the loads P. By well-known graphical methods, these
curves can all be constructed without the use of algebraic processes, with at
least sufficient accuracy for practical computations.
It should be noticed that M is used in two senses : it may signify the
ordinate of the moment curve first mentioned ; or, it may signify a part of the
" moment area " included between that curve and the span and having an
horizontal width of one unit. It is in this last sense that it is used in (2), so
that if it be desirable to estimate the dimension of the terms in any equation,
regard must be had to this fact. It will readily appear in which sense M is
used in the expressions hereafter employed, even though both are found in
the same equation.
Eddy, The Theorem of Three Moments. 29
Let /o = the moment of inertia about its neutral axis of some particular
cross section, which is assumed, as the standard of comparison.
Let i = Iq-t- 1= the ratio of the standard moment of inertia to that at
any other cross section*
Let a, c, a! be three successive piers of a continuous girder of several
unequal spans, in the order of their position, c being the intermediate pier.
Let magnitudes in the right hand span be distinguished from the corres-
ponding magnitudes in the left, by primes.
Let I and t be the length of the spans.
Let t and f be the trigonometrical tangents of the acute angles at c
between the horizontal and the line tangent to the deflection curve.
Lety be the ordinate of any point of the girder above some datum level,
i. e- above the axis of x.
Then, in case of deflection's so small as those occurring in elastic girders,
we have sensibly
D, = y, — y,— lt , iy^ = jf^ — y^—U, .... (3)
in which the deflections are reckoned from the tangent at c to the deflected
girder. Also, the equation of deflections (2) may be written,
DEIo = X{Mis) (4)
in which any ordinate Jf of the moment curve in the span Z, may be regarded
as consisting, in the aggregate, of three parts : viz, Mi dependent upon the
moment Ma at the pier a, M2 caused by the weights P in the span itself, and
Ms dependent upon the moment M^ at c. M^ and M^ are the effect at any
point in the span of the weights P, which are applied to spans other than /.
We therefore obtain by (3) for the two spans under consideration, when the
summation is extended from c, the intermediate pier, to each of the piers
a and cd.
EI, {y.—yc — It) =? 2? i{M, + Jf, + M,) is\
EI, (/a — y« — ttf) = 2-' i{M\ + M', + M',)i':>f]
in which X and ^ are measured respectively from a and «< towards c.
Now the effect of M^^ which is a bending moment at a due to loads at the
left of a, is a moment M^ which uniformly decreases from « to e, as may be
readily seen when we have regard to the fact that M^ is a couple which is
held in equilibrium by its increasing (or decreasing) the vertical reaction at c.
The moment due to a vertical reaction at c increases uniformly from c to
fl, hence
.Miz= Ma{V—x) -T- Z, and M^ = M^ -^ Z, (6)
I .... (5)
30 Eddy, The Theorem of Three Moments.
Let i^ = 2? {Mix) -j- 2? {Mi) , (7)
then is x the distance from a to the centre of gravity of the effective moment
area due to the moments M; x can be found with ease by a graphical process.
Then, from (6) and (7),
^1 = \ i (l—x) xdx -T- \ i {I — x) dx , |
a*3 =: I i srdx -r- I i xds . \
Again, let /, = 2? (3f,0 -^ 2? (Jif,) , • • • (9)
then is /, an average value of i for the moment area due to M^ . Then from
(6) and (9),
h = fj{l-^)dx ^ fAl—x)dx
ix ax -^ J xdx.
In similar manner it may be convenient to let X2 be the distance of the center
of gravity of the effective moment area due to the weights P within the span,
and to let ij denote the average value of i for the same area; but it is not pos-
sible in general to write integrals expressing the values of X2 and i^ in terms
of X, until the distribution of the loading is given : they are to be found in
any case from equations of the form of (7) and (9).
When the value of i varies discontinuously, it is necessary to divide the
limits of the integrals in (8) and (10), so that instead of a single integral we
have the sum of several, each extending over a single portion of the span in
which i varies continuously.
Once more, 2? {M,) = i MJ
2? {M,)
as appears from previous statements, for these are the expressions for the
moment areas due to the moments Ma and Mc respectively.
Equations (5) may now be written
EI^{ya—yc—lt) = «^i'22?(3f2) + U{MaixX^ + M,hx^ 1
Eh{i/a—!/—n)=i\Sf,^t{M',) + U{Mj;,xf, + MA^^^ ^ ^
Let ' t+t'=T (13)
then is Ta known constant, for if the girder is straight at c before deflection,
then T = 0, and in any case T is the acute angle between the spans / and h
when that is small enough to be regarded as sensibly equal to its tangent.
Now divide equations (12) by / and /' respectively and add, then by (13)
z\7A (")
Eddy, The Theorem of Three Moments, 31
This equation expresses the theorem of three moments in its most general form :
the only unknown quantities in it, for a given girder and given loading, are
the moments at the piers a c c^. When there is no constraint at c, then M^ =
M'^; and in any case, M^ — M'^ zz C the couple introduced by the constraint
at c.
The complexity of this formula, as obtained heretofore, has been due to
the fact that the first two terras of the second member have been expressed
in terms of the weights P, and no adequate method has been proposed for
expressing and interpreting the remaining quantities in the second member,
with the exception of the moments.
Let us now derive from (14) the equation expressing the theorem in case
of an unconstrained girder having a uniform cross section, when the two first
terms are stated in terms of the weights P.
In this case i = 1, and we have to determine X2 2? {M^. Suppose that the
area S" {M^ is the sum of parts due to several weights P, then the part due
to a single weight is
M=Pz{l—z)^l, (15)
and this may be taken as the height of a triangular moment area whose base
is / due to the weight P. This triangle whose area = i M2I is the part of the
moment area due to P, and in computing fgS? (ifj) we must find the product
of its area by 'the distance x of its center of gravity from a. Now x=l{l'\-z)
/. xa''c{M,)==kl^,{P{P — z')z-\ . . . • . ,(16)
Also in this case, 5^ = JZ , x^=: II ^
+ MJ + 2M,{l+t) + M'J; , (17)
a well known form.
Again, if the girder is straight at (T, the piers on the same level, and the
cros3 section constant, we have
6 [1^,5, + ^^'2^2] = M^l + 2if,(Z+/0 + MJ; ,
a form of the equation first given by Professor Ohas. E. Greene, 1875, in
which A2 and A'2 are the moment areas due to the applied weights.
SOLUTION OF THE IRREDUCIBLE CASE ;
Or, to express the three roots of the general and complete equation of the third
degree in finite^ algebraic* and really performable functions of the
coefficients oj the equation^ when these roots are all real
and at least one of them is rationaL'\
By GuiDO Weichold, of Zittau^ Saxony.
m
Let a, d, c be the roots of the general and complete equation of the third
degree :
x* + ^a^ + ^ar + (7 = ,- and
6
^-l + V-3^^^-1 V-3^ Whence,
, .; , ^ (2a — ft — c) + (ft — c) V— 3
2 '
f f^h X a {2a — h — c) — {h — c)\f^—^ ,
then from the known relations :
a+ft + c^ — A ; ab -{■ ac -\- be z^ B ; abc = — C:
3 p — p' 2^ + (p + p') ^ + (pi + p'i)
J _ —A + er^ + e^' _ _ d'p, — Of/, ^ _ 25— ^e'pr+ 0p',) ^ _ 3c .
3 B'p — ep' ' 2^ + (^p + 0p') ^+(e'p, + 0p',)'
* From the above enunciation it iH obvious, that the so-called trigonometrical solution of a cubic equation
is no more a solution of the irreducible case than the division of an angle into three equal parts by means of a
protractor, would be a solution of the problem of the trisection of an angle, as it in reality amounts to no more
than the copying of the roots from tables, where their approximate values are indirectly given.
f In the case where the three roots are all irrational, it would be as absurd to require these roots to be deter-
mined with numerical exactness, as it would bo to require a numerically exact root of a number which is an
imperfect power j)f the degree of tjie root to be extracted. Moreover, the determination of the approximate
values of such roots has nothing to do with the problem of the irreducible case, as the operations indicated in
Cardan's formula become performable when treated by approximation (expansion into series).
^Consequently p and p^, as well as pj and p\y are eonjugnte complex functions of a, 6, c, i. e., quantities
consisting of two parts, one of which is real and the other imaginary.
82
Weichold, Solution of the Irreducible Case, 33
_ — ^ + gp + ^p^ __ Opi — O'p'i __2 B— (dpi + e'p'i) 3g
^■" 3 ■" 0p— >p'~ 2^ + (^p+>pO'~ if+(ep, + ^p'i)'
or written collectively :
a
!>{ =
^+^^+^1^', ^jp'-lh- _ 2/?-^(p.-^
367
Up' 2^ + 0} p +
-B +
Now the four auxiliary quantities p, p', p„ p', can be expressed in terms of
the coefficients A, if, C of the proposed equation by means of the following
seven relations :
1°. pp' = dp.fl'p' = d'p.dp' = (a + 0ft + &€) {a-\-n + dc) = (ft + Off + O'c) (ft + 0'a
+ ec) = {c-\-ea + n) {c + e^a+eb)=i { [(« _ j) + (« _ c)]* + 3 (ft — c/}
= i { [(6 _ c) - (a — ft)]^ + 3 (a - cy } = i { [(« - + (* — c)]
+ 3 (a — ft)^}* = (a — ft)* — (a — ft) (o _ c) + (a — c)' = (ft >- c)
+ (6 _ c) (a _ ft) 4. (« _ ft)2 = (a — cy—{a — c) (ft — c) + (ft — c)
= (a* + ft* + c*) — (aft + a« + ftff) = ^^ — 3 5 = N, for brevity ;
2°. p,p', = e^i.&p\ = 0'p,.6p', =(ftc + dab + e-a^?) (ftc + B'ab + dtK?) = (ac + Bab + nc)
{ac + erab + Bbc) = [ab + Bac + B'bc) {ab + 0'fltc + Bbc) = i { [ft (a — c)
+ c (a— ft)]*+ 3a* (ft — cf} = i {[c (a — ft) — a (ft— c)]*+ 3ft* (a— c)*}
. = i{[a(ft_e) + ft (ff_c)]* + 3c*(a — ft)*}* = ft* (« — <?)* — ftc(a — ft)(a-c)
+ c* (a — ft)* = c* (a— ft)* + a<? (a — ft) (ft— c) + a*(A— c)* = a* (ft — c)*
_ aJ (ft _ c) (a — c) + ft* (a — c)* =-a*ft* + aV + ft*c* — aft<? (a + ft + c)
= 5* — 3^C=i^';
3*». pp'i + p'pi = 0p.0'p'i + fl'p'.Opi = d'p.Op'i + Op'.O'p, = (a + Oft + 0'c) (ftc + 0'a<? + flaft)
+ (a + 0'ft + 0<?) (ftc + 0ac + 0'aft) = {c-irBa + Brb) {ah + B^bc + 0ac) + (c +
Bra + Bb) {ab + Bbc + 0'ac) = (ft + 0c + M (ac + 0'a& + 0ftc) + (ft + 0'c + 0a)
(ac + 0aft + 0'ftc) = 6aftc + (a*ft + aft* + a*c + oc* + ft*c + ftc*) (0 + 0") = 9aftc
— (aft + ac + ftc)(a + ft + c) = ^if— 9C=P/
* These expressions show that iV^and N^ can be reduced to the form : a2 + 8/?* = (a-f-/? y^ZZsy(a—p l/--3),
when a, 6, c are real and integral, and to the form : — ^ = — l_i_l L i l_l i ^ when a, 6, e
are real and fractional, a, ^, 7 denoting integers.
9
34 Weichold, Solution of the Irreducible Case.
4°. p^' + p'* = (p + p') (0p + 0'p') (O'p + dp') = {2a^b — c) {2b — a — c) {2c — a— b)
= {{a—b)+{a—c)} {{„ — c) + {b — c^\ {{a — b)—{b — c)}=2{a'-\-b'-\-c')
— 3{ab-\-ac + be) {a-\- b + c) + 27 abc = — 2A' + HAB — 27 C = SP—
2AN=M;
5° p? + p'? = (p, + p'O (0p, + 6'p\) {d'p, + ep\) = {2bc —ab — ac) {2ac — ab — be)
{2ab ~ ac — be) = {b {a — c) + c {a — b)} {a{b — c) —c {a—b)} {a {b —
c)-\-b{a — c)}=2 {aW + (^(f' + ftV) — 3abc {ab + «<? + be) (a + 6 + c) +
27«'ftV = 2ff' — 9ABC+27C'^ = — 3CP + 2BN' = M' ;
6°. p^pi + pV. = (0p)'(0p.) + W{^9'^) = (0'p)'(0'Pi) + m"W^) = {a + Bb + ^cy
{be + d'ab + dac) + (a + d'i + Be) ^ {be + dab + 0'ac) = {e-\-da-\- ny{ab +
O'ac + Qbc) -\- {e -\- &a -\- eb)\ab + 0ac + nc) = {b + 6c'+ 6'ay{ac + He +
0a&) + (i + 0'c + eay{ac + 0i<? + d'aJ) = — abe {a + 6 + c) + 4 (a^i^ + aV
+ 6V) — (o* + ft' + c*) (aft + «c +bc)= — 9A C+ QB" — A'B = 3N'— BN
= AP — 2BN;
7°. pp? + p'p'? = (0p) {Bp,y + (e-po (O'p',)^ = {&p) (0'po^ + (0p') {ef,y = (« + e* + e'c)
(ftc + 0ao 4- e^aby + (a + 0'i + dc) (ftc + C'ac + Baby = (c + 0a + 0'ft) (ai +
Qbc + d'a^?)'' + (c + e-a + 0ft)(aft + He + dac)' = (6 + 0c + e^a){ac + 0'6c +
flaftf -\-{b + &e + da) {ac + 0&c + O'ab)^ = — abe {ab + ac + be) + 4abe (a* +
j^^c') — {a+b + e) {qW + aV + 6^<^) = 97^(7— eAW + AB" = AN' —
3CN=2AN'—BP;
From the foregoing' seven relations follow :
1°. pp', — p'p, = {a-\-Bb + &c){be + &ae + Qab) — {a+n + de){bc + Bac + erab)
= {a — b){a — c){b — e) V"-^ = V(pp', + p'pO' — 4pp'p,p', = VP^— 4iVi\^'
=. ^ V^r3 , putting yl^ ^^'- ^'' = S;
2°. p» — p '» = (p — p') {Bp + O'pO (g' p — Bp') = 3 (a — ft) (a — e) {b — c) V^=l
= V (p^' + p'7 — 4py« = ViVP — 4i\'^ = V"(aP — 2^JV^)^ — 4J^='
= V 9P* — 12^PiV^ + 4A'N^ — 4N^ = V 9P* — 4Jr (iV^^ — ^*ir + O'P)
- = V 9P^ — W.9N' = 3 V T'''2^4WK' = 3SV -^ ; .
3°. p\ —£! = (p i — p'i)(ep i — 0'p',) (0'pi — 0p',) = — 3a&c (a — &)(« — c) (ft — c)
V— 3 = V(p? + p'i*)' ^^^f^ = VM''—4N''= V(^^^3CP+2BWf^^iW'
= V9C-P-— \2BCN'P-\-4IfN'^— 4N^ = VdC'P'—4:N'{N'''—B'JV'-\-3BCP)
= V 9C'P^ —^KdC^N = 3C\^P' — 4NN' = 3C5 V^rg ;
Weichold, Solution of the Irreducible Case. 35
4°. p«p, — p'Y, = (a + eb-\- &cf{bc + d'rtft + 0«c) — {a + Orb + ec)\bc + Bab +
0'«t') = (a -f ft + c) (« — 6X« — c){b — c) V^^Ts = V(7^r+pW^^yp7i
= ^f\Ay— WBNf^^iF^N' = V A'F' — 4AJJFA' + 4:B'J!i' — AN'^JV'
= V A^F^— 4N{j^N'— B'N + ABF) = V A^ P^'^ANTA'^'N'
5°. ppf — p'p7= (« + 06 + 0'^r)(Jc + 0'a6 + 0acf— {a + n + ec){bc + Bah -\- eracf
= (a* + a<? + 6f)(a — 6)(a — c)(6 — c) V — "3 = V "(ppf + p'p'f )' — 4pp'p?p'i'
= V {:2AN' — BPf — 4i\^A^'^ = ^/l^PW'^^^^ABPN' + lfF^^^^Wrif''^
= ^/WF' — AN' {NN'''^~~yPN~+AUr) = V BW^iU'TWN'
= B V P^ —ANJH' = BSV^^ ;
and, moreover :
pp'. = J {(pp', + p'pO + (pp',-p'pO} =M P+ VP^-4iV^'} = J {P+<SfV=3};
p'p,= } {(pp', + p'p,) — (pp', — p'p,)} = i {F—^ ^F'—A^N' } = J {P_<SV— 3};
p' = i {(p^* + p'") + (p' — p'")} = J {if + V IP — 4 A-'} = J {3P — 2AN +
35V^Il3} ;
p" = J {(p» + p'») — (p«— p")} = i{M— VM' — 4JV'} = J {3P—2AN —
p? = } {(pj + p'f) + (p] — p'f)} = i {M' + Vif'^. — 4iy^"} = } { _ 3(?P +
2Pi\^' + 3^-5 V — 3} ;
p'f= i { (pj + p'f) _ (pj _ p'») } = J {M'—VM'-' — 4N"} = J {_3CP +
2BN' — 3CS \^'^3} ;
p^pi = } {(p^p, + p'yo + (p2p, — p'Y,)} = i'{3N'—By+A VP* — 4iVi\^'} =
i {^P — 2PiV + J5 V'^3} ;
p'y, = i {(p*p, + p'y,) — (py — p'y,)} = i {sf'— bn—a v p^ — 4 a*^'}
= J{^P — 2Piy^— ^5V^^3} ;
pp? = J { (pp? + p'p'i^) + (pp? — pyi")} = i {^iT' — 3C2V + if V> — 4yT'} =
J {2JiVr'— PP+ P5V^^} ;
p'p'i* = J {(ppf + p'p'f) — (pp? — p'p'i')} = J {^iT' — 3CN— B V P^ - 4NN'] =
\{2AN'— BP— BS'sf^^] .
Now, as these values of pp'„ p'pi, p', p'', pj, p'f, p'^p,, p'y„ ppj, p'p'f, are com-
plex (see note, page 32) when a, b, c are simultaneously real, as appears from
V4 2{jf' P''
^ > showing that S
36 Weichold, Solution of tlie Irreducible Case.
is real and rational when a, ft, c are real anct rational, the determination of
P> p'j Pi> p'l) ^y the extraction of the cube rooUof p^ == i {M + 3S\/ — 3} ;
p'»=i{Jf— 3aSv'=^} ; pi = i {M'+3CS^^^}; pf = } {Jf _3C»V^^^},
would involve the irreducible case. But the values of the four auxiliary
quantities p, p', pi, p\, can be found without the operation of extracting the
cube root of p^ p'^ p?, p'f, by the (letermination, in various ways, of the factors
respectively common. Thus:
I. to pp' and pp'i, viz. p ; to pip'i and p'pi, viz. p^ ;
PP PPu P J PiPi Ppn Ply
II. to pp' and p^ viz. p ; to pip'j and p?, viz. pi ;
PP P ) P ) PlPl Pl> Pi '
III. to pp' and p^pi, vi/^ p ; to p^p'j and p^pi, viz. pi ;
PP P Pl» P ' PlPl P Pi' Pi »
ly. to pp' arid pp?, viz. p ; to pip'i and ppf, viz. pi ;
PP PPu P > PlPl PP l> Pi >
V. to pp'i and p^pi, viz. p ; to p'pi and p^pi, viz. pi ;
PPl P Pi' P ' PPl P Pl» Pl>
VI, to pp'i and pp?, viz. p ^to p'pi and ppj, viz. p, ;
PPl PP 1> P y PPl PPl) Pl>
VII. to pp'i and p^ viz. p ; to p'pi and pi, viz. pj ;
PPl P J P J PPl P IJ Pu
VIII. to PP? and p^ viz. p ; to pp? and p?, viz. p?, whence pi = \pp? | p? ; ♦
'^p'lp'? " p'^, " p'; " p'p'? '^ p'?, " p'?, " p'^ = .J^fjy^;
IX. to p^pi and p^ viz. p^ whence p = \p2pi | p' ; to p*pi and p?, viz. pi ;
P pi p J p J P — Vp'yi I p" ; to p'yi and p'f, viz. p'l ;
X. to p^pi and pp?, viz. ppi, whence p = *^^ and pi =
2^ 2
p'pi I PPi p'Pi I PPl
" P'y. " p'p'f " P'p'x, " P' = ^rffer " p'l = ^=W
pplppl ppllppl
and satisfying at the same time the above seven relations.
Therefore, by means of the sign ~1 , the roots a, b, c can be symboli-
cally represented as follows :
* The sign . | denotes the factor in question common to the quantities on either side of it.
Weichold, Solution of the Irreducible Case.
37
■^ + l\? + l\9'-^ + l\9P'\p' +
PPiP
'3
3
3
flTPiP' P
?-^|,P.p'>
Pt
H H
p'
^PP
p"
pp'
p'^
25
01p'-0'1p'>_ 2/?-^[p.p'Jp?-^[p.(>'.
pi
3C
2^+^|ppip' + ^[pp'
p
/3
30
Ji +
^ +
l\ p.p'> I p? + ^} p.p'i 1 1
pt
and so on, by using for the values of p, p', pi, p'l the other combinations speci-
fied above for the determination of the common factor in question.
Or, writing the roots in terms of the coefficients A, B, C :
A+l^: A' — 3B
AB —^C+^/[AB —96?— 4(yl- — 'dB)[B' —dA C)
+ ^^ A' — 3B
AB — 9C— A^{AB — 9Cf — 4 {A' — 3B){B' — SAC)
etc.
The determination of the above common factor can be effected, as may-
be seen from the annexed numerical examples, by means of a limited number
of essentially algebraic operations ; that is to say^ p, p', pi, p'l are finite, algebraic
and really performable functions of the coefficients -4, B^ C of the proposed
equation, j. e. f.
Corollary.
The expressions: «=
— pi=pi. h^-
c=-^P'
10
0p-
P
0'p'
0'p'
7'; *=
- ^P'. (p.
^p i— ¥i ^ _ (p.— p'i) +(pi+p 'i)V-
e-p— 0p' (p _ p') + (p + (7) v;
■P'i)-(Pi + P'.)V^:^,
(p — p') — (p + p')V-3
38 Weichold, Solution of the Irreducible Case.
. _^,, _ P(p + p')- (p -pO^V-3 . ^ ._ P(p-pO-( p + p')'^V-3 .
p.+p. — T^— 'p'~p' 2:^^-^ '
(the last two resulting from the combination of: pp' = N\ pip'i = N']
pp'i = — — — - ; p'pi = ^^^ ) furnish, after the elimination of
ff, p + p', p — p', gi + p'„ p, — p'l on the one hand, and of ft, p + p', p — p', pi + p'l,
Pi — p'l on the other, the two equations :
, 2N'+ { P+S) c , _ _ 2N' + {P—S)c .
~" 2Nc + (P— 6') ' ^~ 2Nc + (P + 6') '
or, written collectively, the following formula :
2iV'+ {P^S)c ^
:h
2Nc + {P^ S) '
by means of which, when one root of the proposed equation is known, the
remaining two may be found in terms of the known one and the quantities
N^ N\ P and S, which have already been calculated for the determination of
the first root.
Scholium.
The following considerations will show that algebra itself points to the
foregoing solution of the irreducible case:
I. The elimination of three of the four quantities p, p', pi, p'l, of p', pi, p'l
for instance, from the four simultaneous equations above obtained :
leads to the first equation in p : ■
2Np* — [3P — 2 AN— 35V— 3] p» — N [3P — 2AN-\- 35V— 3]p + 2^ = ;
which becomes decomposable into a product of two factors by putting in the
last term for iV' = py * its equivalent :
3,3 3P— 2^J^^+3-S'V"=l 3P — 2^aV— 36V=1 .
P P = f • 2^ ~~ ' ""^^ •
|2p''— (3P — 2^iV^+35V^r3)| j^j^r^_'^P—2^^—^SV — 3} ^q.
Weichold, Solution of the Irreducible Case. 39
which can be satisfied by putting :
either 2p^— (3P— 2^3^+35V— 3) =0; or JV^p _ ^^'— -?4^~1?:^^-^ = ;
3P—2yLX+3SV — 3 , 3P—2AN—3S\/ — 3 o''
whence p"* = — ' ; and p = --.^ = ^- .
^ 2 ' r 2^V pp'
The value of p^ furnished by the first of these equations cqincides with
that previously found ; the value of p furnished by the second, does not satisfy
the proposed equation, if the quotient of = p'^ divided
hy N z= pp' is taken in the ordinary sense, but if taken in the niore general
sense of the factor common to the diviiUnd and divisor, it furnishes the conjugate
value to. p. This interpretation is vindicated by the fact that, as there does
not yet exist any sign to denote a factor common to two quantities, that
operation cannot manifest itself as the result of other operations, as is the
case with addition, subtraction, etc. It would, therefore, be of advantage to
add to the notation of the elementary operations, a sign to denote such a
common factor, by the adoption of which other important results might
likely be obtained.
II. The fact that, though it is impossible to extract the cube root of the
values of p^ p'^ pi, p'? algeTbraically under finite form, yet the extraction of the
cube root of the products p^ p'\ pj p'?, p"^ p'J, p'^ p?, p^ pj, p'^ p'J, p^ pj, p'^ p'J can be
effected in the most general manner, as the values of these cube roots are
respectively :
pp = 3 ; p,p', = A'; ppi = ^ ; pp, = ;
.2
3.V — Jm+ ASV—3^ „, _ 3N' — BN— AS V— 3
ppi= ^ ;p pi =
2
, AN' — 3CN+liS^/—3 ,,„ AN' — 3CN—BS^/—'d
is another striking intimation that the determination of p, p', pi, p', is not to
be effected in all cases by the extraction of the cube root of the values of
P*» p''j pi' p'li but in certain cases by the decomposition of pp', p,p'j, pp'„ p'p„ p*pj, p'^p'i,
pp,, p'p'? into their factors, i. e., by the foregoing determination of common
factors.
WeITH^'LI*. .S'''"^V'/i **t fh*: Im '^*»ci^^'U Ol'^.
XuMEBiCAL Examples
lUu^fratino tke Solution of the Irreducible Cose.
I.
., ' — 1.;., - _ ~%r — t« » = « •.
A= — !•; : h = 73 : r = — !«».
... = .-: — :M: = y = :i7 : » y. = ^ = 1' ■ ''•• = t> - re: = Ah— 9< ' = P
. — ... ^ '^
>*'
. - . = :'J' — i»J V = .V = 1 II I : _ , = 2/;.V _ :v /' = M = .ji.»',.>4 ;
,. - , ^ 1= :S-Y — 7..V = :VJ'^: ,,: — , , ■ = J.V — :v.V = — ».;i.74:
.■/_::<^ _.s .. . ^. — :. ^f—■^>s'—:i
= ", _ 1 J»-. \ —:',: ^ =- -- = .V>_ IJi; x' — 3 :
-• ' •*
, ^ ,/ _ 3, > .;_ . ^ ^^^.. _ J J .^ , ^ , _ _ J/ _ 3- .>^:_ 3
t r .
•I
= 1*V> — 'nJx'— 3: ^v: = — •^•""— ^•«^'— 3;
5 iK-iii^ re-al &lJ raii-jiiril sh-ws that a, L c are real and raiional. and
iLh: The iireducilOe ea>e «:vours here. Tl^erefL-re j. ^ , ;■:. ^ . caijnoi W deier-
iLiLrd oiLerwi>e iLan Vy liie pr':»ee>^ of *• ^.-viuni^-L ::v.-:-t^.** Bui as ii w^uld
l»r :^»:' ial-.rivii? i^« exLau?; all the possible wiv? ...f deivnniLirig iLese «^uaii-
liiit:^ I'V :iiai process, the f •H.jwii-i: three o.i:iV::ia:i.jriS ire deenied sufficient
X.J :ll'j.stra:e that metbLHi.
D^termifiation ofHf Facior Commoji :
1. To rt/ = 37 and ^^ . = — 179 - 42 \ '^^S' v:z » :
Firsi operation : — 179 — i2 ^ — 3 : 37 = — o — % — 3 — -
37
I'lliiTl-.TT
'— o ' -r. o — 2\ — 3>
= — o- \ — 3- \ — 3 J
37
Weichold, Solution of the Irreducible Case. 41
fonner simplified hypothetical
divisor fonner rem. quotient
second operation : 37 : 6 — 2V — 3 =a + /?V — 3;*
a = 5; |5 = 2; i. e. 37 : 5 — 2V — 3 = 6 + 2V— 3.
Therefore:— 179 + 42 V — 3137 = 5 — 2 V — 3; hence
pp' = (5 — 2V^^) (5 + 2V^3) = (— 5 + 2V^^) (— 5 — 2V^^) and
pp',= (5 — 2V^==^) (— 31— 4V^3) = (— 5 + 2V"^3)(31 + 4V = 3).
To decide whether p:=5 — 2V — 3or = — 5 + 2 V — 3, it suffices to com-
pare the cubes of 5 — 2 V — 3 and — 5 + 2 V^^^^ with the above value of
p» = 55 + 126 V^^ ; thus :
p = — 5 + 2V=^;p' = — 5 — 2V^^;pi = 31— 4V— 3;p', = 31 + 4V^3.
2. To pp'i = — 179 + 42 V^ and p,p', = 1009, viz. p', :
dlv'd div'r hypoth. quot.
First operation: 1009: — 179 + 42 V — 3 = a +• (3 V — 3; a = — -^ ;
42
(3 = — --] in integers approximately a = — 5; /? = _!; remainder : — 12
+ 31 V^^= V^^(31 + 4V^=r3);
former div'r simplf. former rem. hypoth. quot.
second' operation : — 179 + 4 V^^ : 31 + 4 V^^ = a + ^ V^-3 ; a = — 5 ;
/3 = 2; i.e. — 179 + 42V^3: 31 +4 V^=^ = — 5 + 2 V=ll. Therefore:
10091 — 179 + 42 V — 3 = 31+4 V — 3; hence pp', = (31 +4V^=^)(— 5
+ 2 V^=r3) = (_ 31 — 4 V"^=^) (5 — 2 V^^) ;
p,p',= (31 — 4 V=T) (31 + 4 V^^) = (_ 31 + 4 V^=^) (—31 — 4 V^=r3).
From the comparison of the cubes of — 5 + 2 V — 3 and 5 — 2 V — 3
with the value of p' = 55 + 126 V — 3, we have, as before :
p = — 5 + 2V — 3; p'= — 5 — 2V— 3; p, = 31— 4V— 3; p', = 31 + 4V— 3.
*m-\-n\/ — 8^(m-f-w]/ — 8)(/> — q\/ — 8) (w»p+8n^) + (n/) — fnq)\/ — 8 mp-{-Snq /np — niq\
^"+7/^ "" (/>+yi/— 8)(/)— yi/^T^"" p2 + 8^2 ^p2-\.Z^ '^\p'2^S^/^~'^
shows that the quotient of the division of one complex quantity by another is in general likewise a complex
quantity. If, therefore, this quotient be denoted as above by a -|- ^i/ — 8, the values of a and p are expressed
by the two formulae :
mp + Snq ^ np — mq
** = p2 + Sq2 ' ^ == p2 + Zq2 »
by means of which a and p are calculated in tlie above as well as in the following examples.
11
42 Weichold, Solution of the Irreducible Case.
3. To p,p', = 1009 and p'p, = 163 — 672 V — 3, viz. p, :
First operation : 1009 : 163 — 672 V^^ = a + ,3V=^;a= .^ ; /? = ^- ;
loo9 loo"
or approximately in integers a = 1 ; /? = ; remainder : 346 + 672 V — 3
= 6 V^=^ (112 — 47 V^=3) ;
second operation : 163 — 672 V^^ : 1 12 — 47 V=^ = a + /3 V^^ ; a = — ;
19
^ ^ ; or nearly a = 6 ; /? =: — 3 ; remainder : — 43 — 27 V — 3 ;
third operation : 112 — 47V^=^: — 43 — 27i/^^ = a + /?i/^I^3; a = i ;
^ ■=■%', or nearly : a ^ ; j3 := 1 ; remainder : 31 — 4 V — 3 ;
fourth operation :_43 — 27 V— 3:81 — 4V— 3 = a + /? V^=^ ; a = — 1 ;
j3 = — 1; i«.— 43 — 27V;=^:31— 4V^3 = — 1 — V^^.
Consequently : 1009 1 163 — 672 V— 3 = 31 — 4 V— 3 ; whence : p,p'i = (31
— 4 V^3) (31 + 4 V=r3) = (— 31 + 4 V=3) (—31—4 V^Tg) ; and p*p,
= (13 — 20V^^^)(31 — 4V=^) = (— 13 + 20V=^)(— 31 + 4V^=^).
From the comparison of the cubes of 31 — 4 V — 3 and — 31 + 4 V^3
with the above value of pj, we have : pi = 31 — 4 V — 3 ; p'j = 31 + 4 V — 3 ;
and from the division of the above value of pp'i by that of p'l, p = — 5 + 2 V — 3 ;
p' = — 5 — 2i/^=^.
Determination of the roots a, 6, c :
a = - ^ + P + P' _ 16-10 ^2 = - P' - 9'^ = - -^^-3 ^ ^
3 3 p — p' +4v^=^
2^ — (pt + p' .) ^ _ 146 — 62 ^^^ —3C _ —270 ^
2^ + (p + p')' —32 — 10 :B + pi + p'i 73 + 62
,_— ^H:0'p-Hp'_— 2^— (p+p')— (p— p')V^^_32+10— 4V^^ V^^_ .
e'p^ — y, _ _ — (pi — p'l) - (pi + p'l) V=^ ^ _ 8 V^I3 — 62 V^^^S ^ p
0'p_0p' _(p_p')_(p_|.p')^_-3- -4v^+10v^
^_ 2i?-(e'p.+ep', ) ^ _ 4g+(pi+p',)+(p.-p'i)v'^^ ^ _ 292 + 62+24 ^
2A + (0'p + dp') 4^ — (p + p')— (p— p') i/^Ts -64+10+12
3C 6C —640
if+0'p.+Vi 2if— (p,+p',)~(p,— p',)i/^ 146—62—24
=9;
Weichold, Solution of the Irreducible Case. 43
_ — ^+0p+0'p'_ — 2^— (p+p')+(p— p'K-^ _ 32+10+4v'^Z3v/^3'_
C g _ -g _ - -5
^ _ ep,—e'p\ ^ _ -(p.-p'.)+( p.+p'.)v^^3 ^ _ 8v^+62 v;^_ ^ ^
0p_0'p' _(p_p;)+(p+p')^^3 ^4/zr3— i6i/_3
^ _ 2B-{ep,-\-e'p \)^ _ 4^+(p^+ p'.)-(p,— p'O v^^:3 ^ _ 292+62—24 __ ^
2^ + (dp + 0'p') 4J — (p + p') + (p— p') 1/^3 —64+ 10—12
^_ 3C 6(7 —540 _
if+ep,+0'p'i ~ 25— (p.+p'0 + (pj— p'i)i/±r3~ 146—62+24- •
Moreover, any two of the roots can be found from the third by means of the
formula given in the text : ^ | = — -j^^?„^^ ; thus :
10*^ tc 2018-442-5 192 «
1° from c = 5 ; a = — — = ^ 2 : and
74-5-274 96
. 2018 — 274-5 _ . 648 _^^
~ 74-5 — 442 — — 72~ '
2°from6 = 9;a = -2018^^74:9__-448_^ ^^^
749 — 442 224
= _ ^^ — 442 -9 _ _ —I960 _ _
^— 74-9-274 "" 392"—'
qofrn,r.« 9. A 2018 — 442-2 1134 „ ,
3 froma=2; J = - ^j^^__ =_ __ = 9; and
= _ 2018 — 274-2 _ _ 147P _ _
^— 74-2 — 442 ~ ^94 ~
II.
^+il^_79^ + 429_
^140"^ 14''+ 140-"'
J 41 „ 79 ^ 429
pp' = A^-SB=N='^, p.^. = B^-3AC=2P = '^, pp, + p'p.
-AB 9C-P- 572930 pp', - p'p, _ j ^F^'- P^ _ 380292 .
-^^~^^-^-~ 19600' "Tirr-^ 3 -^-19600'
44 Weichold, Solution of the Irreducible Case.
, _ P + ^V— 3_— 286465 + 190146 V— 3 _P — S sf
PPi - 2 ~ ^19600 ' PP^- 2~
286465 — 190146 V — 3
19600
3 _ 3P — 2^i\^+3^V — 3 _ —13398021 + 79861320 V — 3
^ ~ 2 ~ 2744000
,3 _ 3P— 2^i\r— 3i S V— 3 _ —13398021 — 79861320V — 3
^ ~ 2 ~ 2744000
^ _ —82672615 + 2447 17902 V— 3 , ,3 _ ' -^2672615 — 244717902 V — 3
P'~ 2744000 ' P' ~ 2744000
N and N' being fractional and S moreover real and fractional, show that the
three roots of this equation are real, rational and fractional ; therefore the
irreducible case occurs again in this example. From the fact that, when
a, ft, c are real, p, p' and pi, p'j are conjugate complex numbers, and that, when
moreover a, 6, c are rational but fractional p, p' and pj, p'l are of the form
^ ""^ — ^^^ and ^^^ , it follows that N = pp' and N' = pjp'j are
45apable of being brought to the form — ^. , and that consequently the
common denominators of p and p', as well as of pi and p'l, are respectively
equal to the square-roots of the denominators of N and N'^ when reduced
to their lowest terms. Whence the denominator of the values of pp' and pip'i
must be equal to the product of the square-roots of the denominators of N
and N\ Accordingly these denominators may be set aside in the determi-
nation of the quantities p, p', pi, p'l by the process of "common factors,"
and supplied afterwards to the factors common to the numerators of those
quantities.
Determination of the factor common :
, , 333481 , , —286465 + 190146 V ;^^ .
*' PP = 19600 "^^ P^P^ = 19600 '^^^- P-
First operation : — 286465 + 190146 V^=^3 : 333481
-_1+ 47016 + 19014 6 V=r3 _ _ . , « ,370 (31691 - 2612 V^:^)
333481 - ^-^-^^ ^ ^35i8i >
Weichold, Solution of the Irreducible Case. 45
31691
second operation : 333481 : 31691 — 2612 V — 3 = a + /?V — 3; a =
2612
/3 z= ^-Tf^ ; in integers nearly a = 10 ; ^ = 1; remainder : 8735 — 657 V — 3
third operation : 31691 — 2612 V — 3 : 8735 — 557 V — 3 = a + /3V — 3
961 . 461
a = v'^ ; (3 = ~ ; nearly a = 2 ; /? = 1 ; remainder : — 2492 — 205 V— 3
fourth operation : 87a5 — 557 V ^^: — 2492 — 205V — 3 = a + /?V — 3
55 47
ar= — — ; /^ = 7^; nearly a = — 3; /3 = 2; remainder: 29 — 1202V — 3
fifth operation : — 2492 — 205 V — 3 : 29 — 1202 V — 3 = a + /3 V — 3;
2 Q
a = -5 ; /3 = — ~; nearly a = ; /3 = — 1 ; remainder : 1114 — 176 V — 3
13 13
= 2 (557 — 88 V — 3) ;
sixth operation: 29 — 1202V^r3; 557 — 88V^r3 = « + /? V^^; a = l^
/3 = — 2; I. 6. 29 — 1202 V ^=^; 557 — 88 V^^ = 1 — 2 V— Tjj;
therefore • " ^^^^' + ^^^^^ ^ " ^ I ^^^^ - 557-88V-3 ^j^
therefore. - ^^^^^^ - -- ^^^ ^ • Whence
'— (557— 88 V— 3)(557+88V^ _ (— 557+88V— 3) ( — 557— 8 8 V— 3) .
^^~ 140 ' " 140 140 140
, _ (557— 88V— 3)(— 629+242V— 3)_ (— 557+88V— 3 ) ( 629— 242V— 3) .
PP> 140 ilo "■ 140 140
and after a verification similar to that in the preceding example, by comparing
,, , . 557 — 88 V ^=^ , — 557 + 88 V -^ .,, „ 1 ^ «
the cubes of —r^ and -. -^ with the value of p*^
140 140 ^
3P—2AN+SSV — 3 — 133988021 + 79861320 V — 3 „ ,
= ! , we fijid
2 2744000
_ —557 + 88 V — 3 ,_ —557 — 8 8 V — 3 _ 629 + 242 V— 3
P - 140 ' ^ - 14b: ; P' - 140
, _629 — 242V^^
P' - 140 •
Determination of the roots a, b, c :
__^ + p + p'_ — 41 — 2-557 _ _J.r __ pi — p'l 2-24 2 V — £
^ 3 ~ ~ 3140 4 ~ p — p' ~ 2-88 V^^B
12
40 Weichold, Solution of the Irreducible Case.
11 2/y— (p. + p'.) _ _ — 1580 -^2<)29 __li__ BC
~ T~" 2.4T(pT"p'T ~ 82 — 2oo7"'~ 4~ i' + p. + pi
1287 __n,
- — 790 + 2-629 ~ 4 '
—A + e'p + ef,'_ — 2A — (p + p') — (p — p') V — 3 — 82 + 2o5 7 + 6-88
_ _ _ _ _ __ —
_1S gV-^p'i_ — (p.-p'.)— (p.+p'i)V— 3_ — 2-242V— 3— 2-629V— 3
-7~ ^p— 6p' ~ "_(p_p')_(p+pOV-^ - -2-88v^ + 2-oo7v=^
_ i5 _ _ 2Zf_— (d'p, + 6p\) _ _ 4/y + ( p . + p',) + (p, — p'.) V^^~3
7 '2A + {&p + 0p') 4J — (p + p') — (p — p') v^:^
— 3160 + 2-62J> — 6-242 1.1 36'
1«>4 -i- 2-oo7 + 6-88 7 Ji + ^p, + 6p'i
W 6-429 13
2// — (p, + p',) — (p, — p',) V^^ —1580 — 2629 + 6-242 7'
__,4 + 0p+^p'_^2.4 — (p+p')+ (p— p')V^^ _— «2+2-557— 6-88_5
c_ -^ _ -^^ _.-^_ - _ (•.i4(^ -—J
— _ ^P'TL^P'i _ _ — (p>— Pi) + (P> + P'') *^—'^ — _ — ^'242 V— 3 + 2-629 V^
— Bp'—e'p' — —If, -7) '+ (p + p') v_ 3 ~ — Ji-88 V— 3— 2-55fV-^"3
^.?^_25— (0p, + ^p',) ^ _ 47/ + (p, + p',) — ( p. — p',) V — 3
5 2 A + (0p + 0'p') 4J[ — (p + p') + (p — p') V^3
_ _ --3160+2- 629+6-242 ^ 5 _ 3C ^ _ _ 66'
164 + 2-5.57- 6-88' 5 ~.«+0p, + ^p', '2/y-(p,+p',)+"(p7-p',)V=^
_ _ (5;429 _ 3
~ —1580 — 2-629 — 6-242 ~ 5 '
III.
o , 81 ^ 181 , 3 .V
oo 38^) 4 i
J 81 jy 181 .^ 3
55' 385' 77'
^ ^ 16062 _ 11243 4 , ^, _ 7246 , ^ _ 50652 . ^ _ 932 V266
' ~ 21175 ~ 148225' ^ ~ 148225 ' "148225' ~ 148225
, _ 25326 + 4()6 V 2<>6 V — 3 , 25326 — 466 V 266 V —3
P^' - 148225 " ' PP'=^ 148225
3 ^ —34498548+538230 y/2 m V— 3 . ,3 _ —34498548— 538230 V266 V— 3
^ ~ '570<)<)625 5 P- - 5706(3625
3 _ 171856 + 20f)70 V 2(KJ V — 3 . ,3 ^ 171856 — 20970 V 266 V^
^' 570()6625 ; P 1 — 57066625
Weichold, Solution of the Irreducible Case: 47
S being real but irrational shows that a, 6, c are all real, and either all
irrational or one rational and the other two irrational, as appears from the
formula : , > == ^ ^i^- vv. - 1-: i where N, N\ F are rational and S irra-
b) 2Nc + {P^ 8) ' '
tional in the present case.
The composition of JV; iV', P in terms of a, ft, c* shows, moreover, that
in the latter case N and N' are capable of being brought to the form :
irrational factor in S) and indeed :
vr_ , _ 126^ + 3-266-lP _ (126 ± 11 V266 V^^IT) (126 =F H V266 V— 3)
99
385^ 385 385
_ (—126 T 11 V 266 V— 3) ( — 126 ± 11 V2 66 V — 3 ^,_ , _ 8' + 3-266-3^
- 335 ^3g5 ,iv_p,p,_ gggj
- (8±3V266V — 3) (8=F3V2 66 V— 3) _ (— 8 ^ 3 V 2 66 V— 3)
~ " 385 ~ 385 — 385
(— 8±3V266V^ , 126±11V266V-^ — 126=F11V266V^=^.
i - -^ , -= ; whence p= — = — -— or = ■ — jr^^^ >
3y5 ' '^ 386 385
, 126 =F 11 V 266 V— 3 — 126± 11V266 V-3 8±3V266V— 3
P-~ 385 " -"r_— gg^ ,p,_ ^^
— 8=F3V266V— 3 , 8=F3V266V— 3 — 8±3V266V— 3
"^= 385 'P'= 385 ''= 385 "
,. xu u f 126 — 11 V 266 V — 3 126 + 11 V 266 V — 3
Now as the cubes of ;— , — ^ — -^- »
o86 385
8 — 3V266V — 3 — 8 + 3V266V — 3 ., ,. , .,, ,, ,
are identical with the above
385 ■ 385
values of p*, p'', p?, p',', it is evident that these numbers are the values of p, p',
pu p'l) respectively.
Detennination of the roots o, h, c :
_— ^+p + p'_ — 7-81 + 2-126___5 __p, — p',__— 2-3V266V — 3
^~ 3 ~ 3-385" ~~Tl~~ p — p'~~ — 211V266V^3
__3 _ _ 2if — (p, + p'O _ _ 2181 + 2-8 __ 3 __ 3C
~ Tl~ 2^ + (p + p'j" "" 2-7-81 + 2126 ~ Tl~ B+ p, + p'l
— 5 3-3 _ _ ^ .
~ 181 — 2-8 ~ Tl'
* See note, page 88.
4s
Weicholi». .S'hifitjii ot tht Irrt'hicihif Ca.i«e.
b =
H V
/— 3
— A + ^f) + f)p' _ —-2.4 —
3 -—-■
-•21Sl—21'X-\-'2lW^>\^—S\^—:i
6-as3
Hp — Hp- —
'#i.»
j> — ^»— p-e s/_:3 i^lU^JtWv^— 3— 2^12%/— 3
•V- i
' \' r
,' .i> _ 2H — *(fi>:jT Vl)
iJ — i»*V>-rV
4i,'_,p.-p-. ^ p.— p.)v^33 41sl — 2S — 23N'i^\^— 3 \^-^
\Li^_ p~_L~~;~:;, ;'=ii~~4 7si— 2i2»i~2iU'3iiJN'=3\'=3
lis — ax'lS^
:'i — \ '-"'.;■
•;\' r — \'-:>>
3(
33»3— Ux'jiVJ
t5<
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^ _L. jfp, -j- Op',
■» ^"«
r>is
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21
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f =
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-i-^'— ; - » - i~—r y/^li ~ ~4": >1— 2 US— 211m\>3x'~3\
—3
- 1 i
^ - — \ ," ' - \ r — \ >
sc
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18
49
DESIDERATA AND SUGGESTIONS.
By Professor Caylky, Camhrifige, England,
No. L— THE THEORY OF GROUPS.
Substitutions, and (in connexion therewith) groups, have been a good
deal studied; but only a little has been done towards the solution of the
general problem of groups. I give the theory so far as is necessary for the
purpose of pointing out what appears to me to be wanting.
Let a, 3^ ... be functional symbols, each operating upon one and the
same number of letters and producing as its result the same number of
functions of these letters; for instance, a (.r, y, r) = (A', K, Z), where the
capitals denote each of them a given function of (x, //, z).
Such symbols are susceptible of repetition and of combination ; or (x,y^ z)
=. a {X, 1\ Z), or 3a (x, y, z) •=. (i {X. }\ Z), zz in each case three given
functions of (.r, ^, 2), and similarly a'*, d-3, &c.
The symbols are not in general commutative, fx3 not zz 3a ; but they
are associative, af3,y zz a.l3y, each ir a3y, which has thus a determinate
signification.
[The associativencss of such symbols arises from the circumstance that
the definitions of a, /3, y, . . determine the meanings of a3j ay, &c. : if a,
i3, y . . were quasi-quantitative symbols such as the quaternion imaginaries
/, j, kj then a/i^ and j3y might have by definition values <^ and e such that a3.y
and a.jSy ( zz ^y and ae respectively) have unequal values].
Unitv as a functional svmbol denotes that the letters are unaltered,
l{Xj ^, z) zz {x, //, ;:) ; whence la zz al zz a.
The functional symbols 77ir/// be substitutions ; a (x, //, z) zz (^, 2, jr), the
same letters in a different order: substitutions can be represented by the
notation a = ^— , the substitution which changes xyz into yzx^ or as products
XifZ
of cyclical substitutions, a =*^^ ' \ zz (xt/z) (tfic). the product of the cyclical
Xf/Z KW
interchanges x into //, y into r, and z into x ; and n into w, w into u.
60
Cayley, Desiderata and Suggestions.
51
A set of symbols a, /?, y . . . such that the product a^ of each two of
them (in each order, a^ or /?a,) is a symbol of the set, is a group. It is
easily seen that 1 is a symbol of every group, and we may therefore give the
definition in the form that a set of symbols, 1, a, /?, y . . satisfying the fore-
going condition is a group. When the number of the symbols (or terms) is
zz n, then the group is of the nth order ; and each symbol a is such that
a" = 1, so that a group of the order n is, in fact, a group of symbolical nth
roots of unity.
A group is defined by means of the laws of combination of its symbols :
for the statement of these we may either (by the introduction of powers and
products) diminish as much as may be the number of independent functional
symbols, or else, using distinct letters for the several terms of the group,
employ a square diagram as presently mentioned.
Thus in the first mode, a group is 1, /?, /?^, a, a/?, a^ {a? = 1, /3' =
1, a^zz^a); where observe that these conditions imply also a/?^ = /3a :
Or in the second mode calling the same group (1, a, (3, y, S, t), the laws
of combination are given by the square diagram
1
a
^
y
8
e
1
1
a
/?
y
S
e
a
a
1
^
€
8
/?
^
e
8
a
1
y
r
7
S
e
1
a
13
b
h
r
1
1
e
^
a
6
e
m
a
8
y
1
for the symbols (1, a, (3, y, 5, e) are in fact zz (1, a, /?, a/?, /?^, a^).
The general problem is to find all the groups of a given order n ; thus
if n zz 2, the only group is 1, a (a^ zz 1) ; n zz 3, the only group is 1, a, a^
{o? = 1) ; n zz 4, the groups are 1, a, a^ a^ (a^ zz 1), and 1, a, /?, a/? (a^ zz 1,
/3^ zz 1, a/3 zz ^a) ;* n zz 6, there are three groups, a group 1, a, a^ a^ a^, a^
* If fi =i 6, the only group is 1, a, a", a", a* (a* = 1). W. E. 8.
52 Cayley, Desiderata and Suggestions.
(a^ = 1) ; and two groups 1, /?, /?^ a, a/?, a^^ {a? = 1, ^^ = 1), viz: in the first
of these a^ zz /?a ; while in the other of them (that mentioned above) we
have a/if zz (3^a, a/?^ = /?a.
But although the theory as above stated is a general one, including as a
particular case the theory of substitutions, yet the general problem of finding
all the groups of a given order w, is really identical with the apparently less
general problem of finding all- the groups of the same order n, which can be
formed with the substitutions upon n letters ; in fact, referring to the diagram,
it appears that 1, a, /?, y, S, € may be regarded as substitutions performed
upon the six letters 1, a, /?, y, S, f, viz: 1 is the substitution unity which leaves
the order unaltered, a the substitution which changes la/^ySe into alyl^eS, and
so for /?, y, S, €. This, however, does not in any wise show that the best or
easiest mode of treating the general problem is thus to regard it as a problem
of substitutions : and it seems clear that the better course is to consider the
general problem in itself, and to deduce from it the theory of groups of
substitutions.
Gambridob, £6th November^ 1877.
NOTE ON THE THEORY OF ELECTRIC ABSORPTION.
By H. a. Rowland.
In experimenting with Leyden jars, telegraph cables and condensers
of other forms in which there is a solid dielectric, we observe that after
complete discharge a portion of the charge reappears and forms what is
known as the residual charge. This has generally been explained by sup-
posing that a portion of the charge was conducted below the surface of the
dielectric, and that this was afterwards conducted back again to its former
position. But from the ordinary mathematical theory of the subject, no such
consequence can be deduced, and we must conclude that this explanation is
false. Maxwell, in his "Treatise on Electricity and Magnetism,'- vol. 2, chap.
X, has shown that a substance composed of layers of different substances can
have this property. But the theory of the whole subject does not yet seem to
have been given.
Indeed, the general theory would involve us in very complicated mathe-
matics, and our equations would have to apply to non- homogeneous, crystalline
bodies in which Ohm's law was departed from and the specific inductive
capacity w^as not constant; we should, moreover, have to take account of
thermo-electric currents, electrolysis, and electro -magnetic induction. Hence
in this paper I do not propose to do more than to slightly extend the subject
beyond its present state and to give the general method of still further
extending it.
Let us at first, then, take the case of an isotropic body in general, in
which thermo-electric currents and electrolysis do not exist, and on and in
which the changes of currents are so slow that we can omit electro-magnetic
induction. The equations then become *
^ (x ~\ -\- — ( X - -^ +—(x ^l-\ + 47t =
dx V dxJ dy^ dy) dz\ dz/
dx \ dx/ dy ^ dy/ dz \ dzJ dt
* * ^ Maxwell's Treatise, Art. 825.
14 53
64 Rowland, Note on the Theory of Electric Absorption.
in which x is the specific inductive capacity of the substance, k the electric
conductivity, V the potential, p the volume density of the electricity, and t
the time.
The subtraction of one equation from the other gives
(1) ^^4. (log ^) + ^/ ^ (log ^.) +^ ^ (log ^) _ 1 $ _ ^ = 0.
m
To introduce the condition that there shall be no electric absorption, we must
observe that when that phenomenon exists, a charge of electricity appears at
a point where there was no charge before ; in other words, the relative distri-
bution has been changed. Hence, if the relative distribution remains the
same, no electric absorption can take place. Our condition is, then,
l = c,
• p
where c is independent of t, and p and p' are the densities at the points
ar, y, z, an do:', /, /. This gives
dt ^ ~ p'
'C,
where c is a function of t only and not of x, ^, z, and po is the value of p at
the time t = 0. As we have
1 dVdm dV d A k\ . dV d A k\ , dV d
0»^!)+f|(>»«D+tr^.O--D'
m dn dn dx dx ^ x/ dy dy
k
where m zz — and n is a line in the direction of the current at the given
point, equation (1) becomes
1 dV dm 1 rfp 47tp _ ^
m dn dn k dt x ~ '
- r cdt
From equation (2) p z= po e "^ ^ ,
and hence
m dn dn ' " ^k x )
Rowland, Note on the Theory of Electric Absorption. 55
If we denote the strength of current at the point by Sj we have
and
an
1 J n ^^i
1 d?n _ po ^ '
(3) ^ — = + r-%
cm — 4:7im^ dn S
k
this equation (3) gives the value of — z=.m at all points of the body and at
X
all times so that the phenomenon of electric absorption shall not take place
As this equation makes m a function of x^ y, z, S and f, the relation in general
is entirely too complicated to ever apply to physical phenomena, without
some limitation. Firstly then, as c is only an arbitrary function of #, we shall
assume that it is constant ;
cm — 47tm^ dn S *
The most important case is where m is a constant. Then
dm _ ^
dn
and
c = 47im, S=SQ€~''y prrpoe"*^.
In this case, therefore, we see that both the electrification and the currents
«
die away at the rate c. The case where Ohm's law is true and the specific
inductive capacity is constant is included in this case, seeing that when k and
X are both constants their ratio, wi, is constant. But it also includes the cases
where k and x are both the same functions of F, 5, or x, y, z, seeing that their
ratio, m, would be constant in this case also.
When 771 is not constant, the chances are very small against its satisfying
equation (4).
Hence, we mxiy in general conclude^ that electric absorption mil almost cer-^
tainly take place unless the ratio of conductivity to the specific inductive capacity is
constant throughout the body.
This ratio, m, may become a variable in several manners, as follows :
1st manner. — The body may not be homogeneous. This includes the case,
which Maxwell has given, where the dielectric was composed of layers of
different substances.
56 Rowland, Note on the Theory of Electric Absorption.
2d manner. — The body may not obey Ohm's law ; in this case k would be
variable.
3d manner. — The specific inductive capacity, x, may vary with the electric
force .
It is to be noted that the cases of electric absorption which we observe
are mostly those of condensers formed of two planes, or of one cylinder
inside another, as in a telegraph cable. Our theory shows that different
explanations can be given of these two cases.
The case of parallel plates does not admit of being exphiined, except on
the supposition that rn varies in the first manner above given, or in this
manner in combination with the others, for we can only conceive of the con-
ductivity and the specific inductive capacity as being functions of the ordinate
or of the electric force. As the latter is constant for all points between the
plates, m would still be constant although it were a function of the electric
force, and thus electric absorption would not take place.
We may then conclude that in the case of parallel plates, omitting
explanations based on electrolysis or thermo-electric currents, the only expla-
nation that we can give at present is that which depends on the non-homo-
geneity of the body, and is the case which Maxwell has given in the form of
two different materials. Our equations show that the form of layers is not
necessary, but that any departure from homogeneity is sufficient. It is to be
noted that the homogeneity, which we speak of, is electrical homogeneity, and
that a mass of crystals with their axes in different directions ^vould evidently
not be electrically homogeneous and would thus possess the property in
question. In the case of glass it is very possible that this may be the case
and it would certainly be so for ice or any other crystalline substance which
had been melted and cooled.
In the case of hard india rubber, the black color is due to the particles
of carbon, and as other materials are incorporated into it during the proceiss
of manufacture, it is certainly not electrically homogeneous.
As to the ordinary explanation that the electricity penetrates a little
below the surface and then reappears again to form the residual charge, we
see that it is in general entirely false. We could, indeed, form a condenser
in which the surface of the dielectric would be a bettor conductor than the
interior and which would act thus. • But in general, the theory shows that the
action takes place throughout the mass of the dielectric, where that is of a
Rowland, Note on the Theory of Electric Absorption. 57
fine grained structure and apparently homogeneous, as in the case of glass,
and consists of a polarization of every part of the dielectric.
To consider more fully the case of a condenser made of parallel plates,
let us resume our original equations. Without much loss of generality we
can assume a laminated structure of the substance in the direction of the
plane YZ, so that m and V will be only functions of the ordinate x. Our
equations then become
.j. (kip, -if = a.
ax ^ ax/ at
Eliminatinsr o we find
^ dt dx ^ dx ) dx ^ dx / ~ '
dV
Now let us make p =. x — — , and as t and x are independent, we find on
integration,
-T^ {P —Po) + 47t (jmi —po^i) = 0,
at
where 2>o is the value of ^ for some initial value of Xj say at^the surface of the
condenser, and is an arbitrary function of f, seeing that we may vary the
charge at the surface of the body in any arbitrary manner. This equation
establishes p bb a, function of m and t only, and as we have
_ 1 dp
^~ 471 dx^
p will also be a function of these only.
Let us now suppose that at the time ^ = 0, the condenser is charged,
having had no charge before, and let us also suppose that the different strata
of the dielectric are infinitely thin and are placed in the same order and are
of the same thickness at every part of the substance, so that a finite portion
of the substance will have the same properties at every part.
In this case m will be a periodic function of x^ returning to the same
value again and again. As p is a function of this and of t only, at a given
time f, it must return again and again to the same value as we pass through
the substance, indicating a uniform polarized structure throughout the body.
15
58 Rowland, Note on the Theory of Electric Absorption.
This conclusion would have been the same had we not assumed a lami-
nated structure of the dielectric. In all other cases, except that of t\to planes,
electric absorption can take place, as we have before remarked, even in per-
fectly homogeneous bodies, provided that Ohm's law is departed from or that
the electric induction is not proportional to the electric force, as well as in
non-homogeneous bodies. But where the body is thus homogeneous, electric
absorption is not due to a uniform polarization, but to distinct regions of
positive and negative electrification.
In the whole of the investigation thus far we have sought for the means
of explaining the phenomenon solely by means of the known laws of electric
induction and conduction. But many of the phenomena of electric absorption
indicate electrolytic action, and it is possible that in many cases this is the
cause of the phenomenon. The only object of this note is to partially gen-
eralize Maxwell's explanation, leaving the electrolytic and other theories for
the future.
ESPOSIZIONE DEL METODO DEI MINIMI QUADRATI.
Per Annibale Ferrero, Tenente Colonnello di State Maggiore^ ec. Firenze^ 1876.
By Charles S. Peirce, New York.
Recent discussions in this country, of the literature of the method of
Least Squares, have passed by without mention the views of the accom-
plished chief of the geodetical division of the Italian Survey, as set forth m
the work above cited, which was first published, in part, in 1871. The sub-
ject is here, for the first time, in my opinion, set upon its true and simple
basis ; at all events the view here taken is far more worthy of attention than,
most of the proposed proofs of the method.
Lieut. Col. Ferrero begins by considering the principles of the arithmetical
mean. A quantity having been directly observed, a number of times, inde-
pendently, and under like circumstances, the value which might be inferred
from the observations is, in the first place, a symmetrical function of the
observed quantities ; for, if the observations are independent, the order of
their occurrences is of no consequence, and the circumstances under which
they are taken, differ in no assignable respect, except that of being taken at
diflFerent times. In the second place, the value inferred must be such a func-
tion of the values observed, that when the latter are all equal, the former
reduces to this common value. The author calls functions having these two
properties, (1st, that of being symmetrical with respect to all the variables,
and 2d, that of reducing to the common value of the variables when these are
all equal,) means. There is a whole class of functions of this sort, such as the
arithmetic mean, the geometrical mean, the arithmetic- geometrical mean of
Gauss, the quadratic mean,* and many others instanced in the text. It i&
shown, without difficulty, that these means are continuous functions, and that
their value is intermediate between the extreme values of the diflFerent
variables, when the latter do not diflTer greatly.
Let (/, (f^ (f\ etc. denote the values given by the observations. Let n
denote the number of the observations; let p denote the arithmetical mean ;.
•This seems the appropriate name for -^ — - .
59
60 Peirce, Esposizione del Metodo dei Minimi Qimdrati.
m
tind let y, jf\ jf'y etc. denote the excess of the observed values over the arith-
metical meaa. Then write
V = f ((/, cr, er, etc.)
for any mean of the observations. Develop this function according to powers
of- jf, 3f, y, etc. We have
dV
— f iP^P^P^ etc.) + — {of + jr + 3(r + etc.) + A ;
where A denotes the terms of higher orders.
Since j:' + a*" + ^ + etc. =z 0,
and / (jp, i>, p, etc.) = p,
this reduces to
F = jp + A.
In considering the value of A, we may limit ourselves to terms of the second
order. As the partial differentials of any species and order, relatively to
o', 0", (T, etc. all become equal when ^r', nf^ of*^ etc. vanish, we may write
d^V_i^V_^^_ ^ ^o
dxf^ ~ dd'^~ dxf''^~^ '~^
. — — — ^^ — /y
d(/.d(f dcTJo" ' '
then
A = J /iJ {sf^ + or"* + /"* + etc.) + y (.r'.r" + ,rfoir + etc.) .
But the square of for] = 0, gives
so that
where k is a quantity which does not increase indefinitely with ti. Now,
when the observations are good, t_J is not large, and, therefore, in such a
n
case no mean will differ very much from the arithmetical mean. The latter,
being the simplest to deal with, may therefore be used without great disad-
vantage. Such is, according to Colonel Ferrero, the utmost defence of the
principle which can be made to cover all the cases in which it is usual to
employ the method ; and all further defence of it is more or less limited in its
application.
Peirce, Esfposizione del Metodo dei Minimi Quadrati. 61
In very many cases, however, it is easy to see that either in regard to
the quantity directly observed, or in regard to some function of it, the zero of
the scale of measurement, and the unit of the same scale, are both arbitrary.
For instance, in photometric observations, this is true of the logarithm of the
light. In such cases, considering such function to be the observed quantity,
we have there two principles, first proposed, in connection with a really
superfluous third one, by Schiaparelli!
1st. The mean to be adopted must be such that if each observed value is
multiplied by any constant, the result is increased in the same ratio.
2d. The mean to be adopted must be one which is increased by a constant
0, when each observed value is increased by the same constant.
Our author's treatment of these principles is exceedingly neat. Using
the same notation as above, write
V = p + A2 + As. . . + A^. . .
where A„ is the sum of the terms of the order n in of, of, of\ etc. The general
term A^ is, therefore, of the form A^ = a^oT + ^Xaf^'-'af + yS^"- V'^ . .
+ ^2j^''a:^V. . •• where 2 expresses the symmetrical sum of similar terms.
In the general term r + 5 + ^ + ^tc. =, n. Since ^ is evidently a function
of ^, we may put ^ zz ^{p)^ and it remains to find the form of this function.
Multiplying every by c, ^ is changed to rp, x to cx^ and the general
term ^Sor^o^' V' etc. = ^ {p) ^x^'x^'x"''' etc. is changed to fp (cp) d^^xTxT^xT' etc.
Since, therefore, F is changed to cF, we have ^{cp)^^ = ^{p)c. Putting
^ z= 1, ^(c) = ^{. Denoting this numerator by ^1, the general term becomes
c
An = -i-i [a,2x» + . . . + ^,lx"jf"r". .. + ...],
Jr
where a, ^, etc., are numerical coefficients independent of p. From this cir-
cumstance it follows that the quantity in square brackets, which may be called
A'^ , does not change when the same constant quantity k is added to all the
observed quantities o', (f, (J% etc. ; for such an addition only increases p by this
same constant, and leaves xf^ ^, xT^ etc., unchanged. Thus the mean in ques-
tion, which may now be written
r = i> + ^^ + ^» + etc.,
becomes, in consequence of such an addition,
P + fc {p-\- kf
62 Peirce, Esposizione del Metodo dei Mimini Qimdrati.
But by principle No. 2, it becomes,
-^'2 I -^^'3
P P
So that, A'2 = A'n =. etc. = 0, and we have
or the arithmetical mean is the only one which conforms to the given
conditions.
Another still more special case, is that contemplated by the demon-
strations of Laplace, Poisson, Hagen, Crofton, etc. It is treated by our
author, but need not be considered in this notice.
It may be of interest to see how Colonel Ferrero is able, without basing
least squares expressly upon the theory of probabilities, to derive the for-
mula for finding mean error. Using always the same notation, he terms
^ n
the mean residual of the observations.
Suppose, then, that there be an indefinitely great series of series of obser-
vations of the same quantity, each lesser series consisting of n observations,
and each having the same mean residual. Then, there being an infinite
number of such series, the mean of their mean results may be taken as the
true value, by definition. For the ultimate result of indefinitely continued
observation is all that we aim at in sciences of observation. Then the num-
ber of the lesser series being j, the result will be
r=M.
Adopt the notation
h = p — V hi = jpi — V ^2 =pi — F, etc.,
then 5, 5i, h^^ etc., are the true errors of p^ pi, /?2> ©te- Let /o» ^o? ^oi etc. be the
true errors of the first series of observations, ^i, y\^ f\ etc. those of the second
series, and so for the others. We have, then, y = o — F= o — p + 5 = a: + 5.
Squaring and summing for the nq values of y, we have
2/ = 2.r2 + 25' + 22ar2S
or, since ^x ■=. 0, and 25 = 0,
2/ = So:' + 25^
Now if Yi be the quadratic mean of the error of ^, we have 25' = nqri^^ and
2y^ =: nqiv? + nqyi^^
Peirce, Esposizione del Metodo dei Minimi Quadrati, 63
or the mean error ^ of an observation is given by
u? = -y- =1 m^ + ri^ .
nq
But it is easily shown (from the equality of positive and negative errors) that
2 ^
>7^ r= i--
n
whence
^n — 1
With regard to the mode of passing from the principle of the arithmetical
mean to the general method of least squares, the best way seems to be first to
prove that the solution Of the equations
UiX = Til
etc.,
is a: = LJ. This is easy, after the rule for the error of a mean is established.
Then, having given the equations
a^x + b^y + CiZ + etc. = ih
UiX + ^2^ + ^2^ + etc. = 722 ;
first, consider these as similar to the equations just given ; thus,
a^x = Til — biff — CiZ — etc.,
a^ = 712 — ^2^ — P2^ — etc.,
etc.,
whence we obtain the first normal equation,
_ [«^i] — [pJ>\ y — [^ ^ — etc.
and the others in a similar way.
The treatise of Colonel Ferrero may be recommended to those desirous of
having a thorough practical acquaintance with the method, as decidedly the
best and clearest on the subject.
ON AN APPLICATION OF THE NEW ATOMIC THEORY TO THE
GRAPHICAL REPRESENTATION OF THE INVARIANTS
AND COVARIANTS OF BINARY QUANTICS,—
WITH THREE APPENDICES.
By J. J. Sylvester.
By the new Atomic Theory I mean that sublime invention of Kekule
which stands to the old in a somewhat similar relation as the Astronomy of
Kepler to Ptolemy's, or the System of Nature of Darwin to that of Lin-
naeus ; — like the latter it lies outside of the immediate sphere of energetics,
basing its laws on pure relations of form, and like the former as perfected
by Newton, these laws admit of exact arithmetical definitions.
Casting about, as I lay awake in bed one night, to discover some means
of conveying an intelligible conception of the objects of modern algebra to a
mixed society, mainly composed of physicists, chemists and biologists, inter-
spersed only with a few mathematicians, to which I stood engaged to give
some account of my recent researches in this subject of my predilection, and
impressed as I had long been with a feeling of affinity if not identity of
object between the inquiry into compound radicals and the search for "Grund-
formen " or irreducible invariants, I was agreeably surprised to find, of a
sudden, distinctly pictured on my mental retina a chemico-graphical image
serving to embody and illustrate the relations of these derived algebraical
forms to their primitives and to each other which would perfectly accomplish
the object I had in view, as I will now proceed to explain.
To those unacquainted with the laws of atomicity I recommend Dr.
Frankland's Lecture Notes for Chemical Students, vols. 1 and 2, London (Van
Voorst), a perfect storehouse of information on the subject arranged in the most
handy order and put together and explained with true scientific accuracy and
precision. On the algebraical side of the subject my readers may consult Sal-
mon's "Lessons on Higher Algebra," Clebsch's ''Binaren Formen" or Faa
de Bruno's treatise more elementary than the former, "Sur les formes
binaires" (Turin, 1876). I propose also to run a course of articles on the
Invariative Theory, beginning from the beginning, through the pages of this
Sylvester, On an Applkatian of the New Atomic Theory^ d-c. (>o
Journal, from my own particular point of view which will be found, I hope,
considerably to simplify the subject.
Any binary quantic may be denoted by a single letter with a number
attached corresponding to its degree, and may therefore be adumbrated by
a chemical symbol with corresponding valence. Thus hydrogen, chlorine,
bromine, or potassium will serve to denote so many distinct binary linear
forms; oxygen, zinc, magnesium, etc., binary quadrics; boron, gold, thallium,
cubics; carbon, lead, silicon, tin, quartics ; nitrogen, phosphorus, arsenic,
antimony, etc., quintics; sulphur, iron, cobalt, nickel, etc., sextics. The sixth
appears to be the highest degree of valency at present recognizable in natural
substances.
The fiictors of any algebraical form may be regarded as in some sense
the analogues of the rays of atomicity in the equi-valent chemical atom —
these rays being w^hat Dr. Frankland, according to his nomenclature, would
have to designate as free bonds ; such rays between two consecutive atoms in
a molecule are conceived as blending in some manner so as to represent some
unknown kind of special relation existing between them ; they may then with
propriety be called bonds or lines of connexion.
An invariant of a form or system of algebraical forms must thus repre-
sent a saturated system of atoms in which the rays of all the atoms are
connected into bonds. Thus, ex. gr. O2 (oxygen combined with itself) will
represent a quadratic invariant of a quadric. Its graph is seen in Fig. 1, (a).
Potash, a combination of potassium, oxygen and hydrogen, having for its
graph that of Fig. 2, will represent the invariant to a system of one quad-
ratic and two linear forms which is linear in each set of coefficients. This is in
fact the Connective between the given quadratic and another obtained by tak-
ing the product of the two linear forms. Phosphorus and arsenic are quinqui-
valent, but form ** tetratomic molecules." An isolated element of phosphorus
may possibly, therefore, be represented by the graph of Fig. 3, which will
correspond, if the figure is indecomposable (which requires examination to
determine), to the quart-invariant of a quintic, and the same for arsenic. So
too the graph to nitric anhydride (Fig. 4) may possibly serve to express the
resultant of a binary quadric and quintic, or this blended with any other
invariant of the system included under the same type [10: 5, 2; 2, 5].*
* 10 is the weight ; 5, 2 the degree and order in the coeflieients of tlie quintic ; 2, 6 the dei^ree and order in.
the coefficients of the quadric. See p. 67.
17
66 Sylvester, On an Application of the New Atomic Theory^ &c.
And in general, the Jacobian to any two quantics will be completely
expressed by their two corresponding atoms 'connected by a pair of bonds.
Nitric acid has for its. graph that of Fig. 5. This will correspond to an
invariant of a quintic, quartic and linear form of the first order in the
coefficients of each extreme and of the third order in those of the middle
form. Such an invariant as is well known, (by virtue of a general principle
about to be stated) is, in substance, the same thing as a lineo-cubic linear
covariant of a quintic and (juadric. The general arithmetical rule (also here-
after to be set forth) for determining the number of asyzygetic derivatives of a
given type, enables us to see that such a covariant exists and is monadelphic.
It may readily be obtained by making the given quintic (after substituting
// f1
—- and — for a: and y respectively) operate on the cube of the given
ay ax
quadratic.
The general principle above referred to, which is extremely easily proved
from the partial differential equation, (but which I believe I was the
first to enunciate) is that every covariant of one quantic or several simul-
taneous quantics may be transformed into an invariant of the same quantic
or set of quantics enlarged by the addition thereto of one additional linear
form ; the degree in the variables becoming replaced by the order in the new
set of coefficients, and the orders in the original sets of coefficients remaining
unchanged.
Thus, covariants might altogether be dispensed with and invariants alone
made the object of study. But algebraists have found and will continue to
find it more convenient to dispense with the additional linear form and to
retain in use covariants as well as invariants. With me, covariants are to be
regarded as simple emanations, so to say, from differentiants which are func-
tions of the coefficients alone, and of which invariants are merely a particular
species satisfying a certain condition of maximum ; this is why the properties
of invariants can with difficulty be made out so long as they are studied
alone ; it was only by contemplating the whole group of differentiants
simultaneously, that I was enabled, after a suspense of more than a quarter
of a century, to set on an irrefragible basis Professor Cayley's fundamental
arithmetical theorem for calculating the number of asyzygetic invariants and
covariants to a given quantic, and also the more general theorem which I
have shown applies to a system of quantics.*
* The demonstration is given in a paper inserted in the Philosophical Magazine for March of this year.
Sylvester, Oyi an Aiyplication of the New Atomic Theory^ &c. 67
I will here give this rule, as it may be useful to us iu the sequel. First,
for A single quantic. — Let i be its degree, j the order of any covariant, w its
weight (i. e. the weight of its root-diiferentiant) . Then we may call its type
\w : i, j]. Now let us, in general, employ (m: i, /) to signify tlie number of
ways in which m can be made up with / parts of which each is either 0, 1, 2, 3
etc. up to /, and let us use the symbol A {m : i^j) to denote {m: i^j) — {m — l :
i,j) ; then A {w : /, j) is the number of arbitrary numerical parameters in the
most general covariant or invariant answering to the type [w: i, j]. It is a
known theorem in partitions of numbers that (m : i,j) = {m: j, i), from which
it follows that the number of arbitrary parameters remains unaltered when the
degree of the primitive and the order of the derivative are interchanged. It is
sometimes more convenient to use the degree of the derivative in lieu of the
weight to express its type ; let then e be the degree, so that € = 2ij — w; then
I shall employ, when desirable, [i,j : e] to signify the same thing as [iv : ^,J^].
If there be several quantics, the type may be expressed in like manner by
[w: ijj; i\j\ etc.], or by [*, J; /', /; etc.: t]. The rule for finding the num-
ber of independent parameters, or the most general covariant or invariant
corresponding to either of these types, then becomes as follows. Let {m: i^j\
i\j'\ etc.) denote the number of ways in which m can be made up of J ele-
ments each comprised between and i, 'combined with / elements each
comprised between and i\ and so on, and let A(m: i, j; i\j] etc.) denote
{m: ijj; i',/; etc.) — (m — l: /, J; i', /; etc.). The number of parameters in
question is A{w: i^j; i', /; etc.) and I may observe that the value of A
remains unaltered when any one i is interchanged with the corresponding J,
and consequently when any number of i's are interchanged, each respectively
with its corresponding j. This theorem of reciprocity for a single quantic is
due to M. Hermite. The above statement, applicable to a quantic system,
constitutes a notable and important generalization of it. In Note D to
Appendix 2, it will be shown that this theorem still further generalized by
employing the method of Emanation (virtually the same thing as Regnault's
law of substitution) admits of the following simple chemico-algebraical state-
ment. In an algebraical compound {in an algebraical sense) m n-valent atoms may
be replaced by n m-valent ones. But it should be observed that this replace-
ment involves an entire reconstruction of the representative graph and con-
veys the notion of respondence or contraposition rather than similarity of
type. (See Appendix 2.)
68 Sylvester. On an Applkatiou of the Xew Atomic Theory, dr.
It may be well here (as it will be useful in the sequel i to soy a few words
more on these differentiants in their relation to covariants. Everv covariant
mav be reirarded as arisinjr from either of two differentiants. as from a root.
One, the coefficient of the highest power of .r, is called a diflferentiant in .r;
the other, the coefficient of the highest power of ^, a differentiant in //. It
is not. fur ordinary purposes such as j»resent themselves in this study, requi-
site to consider more than one of these at a time, and for erreater brevity it
will Ix* understood that, unless I give notice to the contrar}% a differentiant
will always be understood to mean one in .r. I shall also suppose, when deal-
ing with a single binary quantic, that the successive coefficients beginning
with the highest power of .r, are a^ //, r. .... A, k\ J multiplied successively
by the binomial coefficients [»roper to the degree of the form. A differentiant,
Ij. mav then be detined as a rational intei?er function of the coefficients of
equal weight in all its terms in respect to either variable subject to satisfy
the equation
An invariant again may be regarded as a rational integer isobaric function of
the coefficients which is a differentiant both in regard to .r and //, but it may
Vie l>est defined as a differentiant (meaning in one of the variables as j-,) to a
given form or form-system whose weight (in respect of the selected variable) is
the greatest possible that its order in the coefficients admits of. [The double-
ness of the character and the symmetry, direct or skew, of a differentiant
satisfvino: this condition of maximum then be^rome matter of deduction from
the definition.] To each covariant corresponds but one differentiant (in a
given variable) and vice versa, to each differentiant will correspond only one
covariant. In fact, i> being the differentiant in j-, the covariant taking
its rise in L is
where fl- represents the operator,
\ 7Z- + ^^ "/I + '^^ /"+••• ) ^^ ^ belongs to a simple quantic, and
i ^/ -^ + 2X — + . . . . j if it belongs to a quantic system, and where e
is Ij — 2tv for a single quantic, and lij — 2w for a quantic system, / repre-
senting the degree of any one form in the variables, j the order of the
Sylvester, On an Application of the New Atomic Theory^ &c. 69
differentiant in the corresponding set of coeflBcients, and w the weight of the
differentiant. As e can nev^er become negative, we see that the maximum
value of to, when each i and its corresponding j is given, will be -^ for one
form, and J 2^; for a form system. By the weight of any covariant I shall
understand the weight of the differentiant in which it may be regarded as
originating. Precisely as algebraists find their advantage in using covariants
when invariants alone might be made to suflBce, chemists find theirs in the
use of organic or inorganic compound radicals, as unsaturated forms capable
of becoming saturated by the addition of the right number of monad elements
to the unsatisfied atoms, i. e. those through which a sufficient number of bonds
do not pass to exhaust their valency. Thus ex. gr , Hydroxyl H — — is the
linear covariant of the quadratic form oxygen, and the linear form hydrogen ;
this, combined with the linear form potassium, expresses the invariant
potash denoted by H — — K.
As the free valence of a single atom corresponds to the degree-of a single
quantic, so the free valence of a molecule formed by an aggregate of atoms
will express the degree of the corresponding covariant. Let us understand
by the toti-valence of a molecule the sum of the absolute valences of the sepa-
rate atoms of which it is composed. This toti-valence will obviously corres-
pond to the sum, 2i;, above mentioned. Since every bond or connecting line
in the graph passes through two atoms, this toti-valence must be equal to the
free valence of the molecules increased by twice the number of bonds ; but
2/; is the toti-valence, and e (the degree of the covariant) is the number of
unsatisfied bonds, and we have already stated in effect that e increased by
twice the weight of the root differentiant (which for brevity we call the weight
of the covariant) is equal to l,ij ; hence the weight of a covariant (meaning
that of its root differentiant), represented by any chemicograph, is the number
of bonds or connecting lines between the atoms.
Let us consider an invariant or a covariant belonging to a type containing
only one numerical paran^eter, which I shall call a monadelphic form.* Then
this is either decomposable into factors or not ; in the former case it may be
termed composite, in the latter case prime. When prime its graph will also
♦The type itself may also be termed a monadelphic type: so I shall speak when necessary of diadclphic,
triadelphic, etc. types and designate any forms contained under such types as diadelphic, triadelphic, etc. forms.
A family comprising many brothers, or any member of such a family, may each without doing violence to the
laws or usage of language be termed polyadelphie.
18
7() SylvI':ster, (h) an yipplication of the New Atomic Theory^ &c.
be prime, when composite its graph will be composite in a sense which will
be made more clear by one or two examples Let us take as a first example
a graph composed of four triadic atoms of the same name, as in Fig. 6, where
each atom, for instance, represents boron and in ordinary chemical symbolism
would be denot43d by the same letter B^ but w^here for facility of reference I
use four different letters to mark the positions of the several atoms. This
corresponds to a covariant of a cubic for which the complete type, if we use
the weight or number of bonds, is [4 : 3, 4], or, if we use the free valency, is
[3, 4:4]. Xow for a cubic the fundamental types, expressed in terms of the
order and degree alone, omitting the constant number 3, which refers to the
given degree, are 1 . 3
4.0
2.2
3.3.
Consequently, there is but one covariant corresponding to the given graph,
and that is the product of the primitive by the covariant whose order and
degree are each 3, the well known skew covariant (a, ft,^?, (l\x^yy whose
TfnA or liase is the differentiant a(P — 'dabc + 2¥.
It must be well understood that the bonds are not rigid, but capable of
being curved or bent into any desired form. In this case the mode of decom-
position i.s ?elf-evident ; for the skew covariant is represented by the triangle
of Fig. 7, and we have only to draw out the elastic bond AC into the position
A DC and place the atom D anywhere upon it to obtain the given graph. On
the contrary the skew covariant itself is indecomposable and its graph ABC
ia obviously so too. Now let us consider the graph of Fig. 8. If the atoms
at the angles are all triadic, there is no free valency, and the figure represents
the invariant to a cubic form corresponding to 4. in the above table. It will
b<^ found, on trial, impossible to decompose it. But now suppose the atoms
t// ^fh tetradic, the graph will represent a covariant of the fourth order and of
the fourth degree to a quartic, each atom having one degree of valency
nufkfkii'AfiftA. The fundamental derivatives of a quartic, of which all others
Are skXiif^frnicBX combinations, are represented in the following table of order
an/l degree; 1.4
2.0
3.0
2.4
3.3.
SylvesTTer, On an Application of the New Atomic Theory^ &c. 71
The complete covariant answering to the graph will therefore be Till + (iV^
where, X, ^ being arbitrary numbers, U is the product of the primitive (1 .4)
by the cubinvariant 3 . 0, and V the product of the Hessian 2 . 4 by the quad-
rinvariant 2.0. Since, on making either ;i = or ^t^ = the covariant
breaks up and in two different ways into factors, we ought to expect that the
graph should be capable of two corresponding modes of decomposition, and
such we shall easily see is the case. For 1°, the invariant 3.0 may be repre-
sented by the graph of Fig. 9. Now imagine the three points E^ F, G to
come together and blend at I) and at D place a fourth atom. The given
graph is thus recovered. Observe that this could not be done for the case of
triads (corresponding to a cubic form) because, in the figure last referred to,
the valence at each atom A, B, C is quadrivalent. Next, for the decomposi-
tion corresponding to the case of X = where the covariant breaks up into
2.0 multiplied by 2.4, the decomposition will be more easily followed by
considering the graph to be pulled out into the form seen in Fig. 10. We
may conceive this as the superposition of two carbon graphs, one in which the
carbon atoms are at A and B connected by the four bonds AB, ACB^ BI)A^
ACDB denoting the quadrin variant, and another in which the carbon atoms
Cj D are connected by the two bonds CAD^ CBD^ leaving two degrees of
valence free at each atom and thus representing the quadro-quart invariant
or Hessian of the primitive.
I will now pass to the very interesting case which corresponds to one of
the proposed graphs for benzole (or rather for the compound radical obtained
by striking off its hydrogen atoms), a sextivalent hexad molecule of carbon —
not the one proposed by Kekule and which I believe still commands the
general assent of chemists, but that suggested by Ladenburg* and put by him
under the form of a wedge or prism. As, however, the question is one purely
of colligation or linkage in the abstract, it is sufficiently described as a hexagon
in which the three pairs of opposite angles are joined, or, if we please, as two
triangles in which each angle of one is connected with a corresponding angle
of the other. In regard of the atomicity theory, all these modes of colligation
are identical, and the supposition that there is any real difference between
them, or that figures in space are distinguishable from figures in a plane (as I
heard suggested might be the case by a high authority at a meeting of the
* BerichU der deutaehen ehemiaehen Oesellaehaft^ 1869 ^ I4I. I am indebted for this reference to my able
colleague, Professor Ira Eemsen.
72
Stlvester, On an Application of the New Atomic Theory^ &c.
British Association for the Advancement of Science, where I happened to be
present), is a departure from the cautious philosophical views embodied in the
the«>rv as it came from the hands of its illustrious authors and continued to be
maintained by their sober-minded successors and coadjutors, and affords an
instructive instance of the tendency of the human mind to the worship, as
if of self-aubsistent realities, of the symbols of its own creation.
The order (or number of atoms) being 6 and the unexhausted valences
M»ne at each atom) also 6, we must turn to our table of fundamental deriva-
tives to the quartic and shall find that the combination 6.6 is not amongst
them, but that it can be obtained, and in only one way, by composition of the
combinations therein contained. It is, in fact, the product of the cubic inva-
riant 3 . by the skew covariant 3 . 6, which has the very same root a^d — Zabc
+ 2¥ as the skew covariant to the cubic and accordingly has the same graph,
namely a simple triangle. (It may be well to remark here incidentally, that
it follows as an immediate consequence from the conditioning partial differ-
ential equation, that a root-differentiant to any quantic or system of quantics
of given degree or degrees remains such to every other system in which one
or more of those degrees is augmented.) On the other hand the cubic
invariant has for its graph a triangle in which each line is doubled or looped.
I shall show that Ladenburg's graph for the radical to benzole may be
obtained by the superposition of these two forms. Let ABCy^a represent
a sextivalent tetradic hexad (Fig. 11); ABC, with the three loops AayC\
CyiB, Bi3aAj will represent a saturated triple atom of carbon, or the cub-
invariant of a binary quartic. Again, ay(S taken alone will represent a sexti-
valent compound atom, or the fundamental skew covariant of the quartic, and
the superposition of the two figures obviously gives the graph as it stands.
Another form of the product of the same two graphs :would be a triangle
inscribed in another, as in Fig. 12. Here a/i^y, as before, is the sexivalent
molecule and ^7/6' with the additional bonds AliC\ ByA, CaB^ the saturated
one.
A f^irnple hexagon of triadic atoms (Fig. 13) being sextivalent will serve to
r^il^rf:^:ut a derivative from a cubic of the sixth order and sixth degree. Such
ik ^y/ variant, in its most general form, will contain two parameters and be
r^::prfni^rnUifl by a f/^ + u P where U is the Hessian 2 . 2 and V the skew cube
<«:>'yvarjariit ti/4f and it is easy to see that this figure may be decomposed either
j'^Vy ti WvaUrrit^ or 2 trivalent graphs. Thus AB, CD, EF, with the additional
Sylvester, On an Application of the New Atomio Theory^ &c. 73
bonds BCDEFA, DEFABC, FABCDE, will represent the former; two atom
groups such as A, C, E (with the bonds ABC, AFEDC, ODE, CBAFE, EFA,
EDCBA) and B, D, F (with the bonds BCD, BAFED, DEF, DCBAF, FAB,
FELCB) the other. The first method of regarding the hexagon as a combi-
nation of three dyads may perhaps be admitted to throw some light on what
Dr. Frankland styles the two distinct molecular weights of sulphur. When
two atoms of sulphur, regarded as bivalent, are combined by two loops, w^e
have a representation of an isolated element of it as *'a diatomic molecule.''
When three of these letters, regarded now as submolecules, are combined, or
multiplied together into the hexagon, we have a representation of the isolated
element as " a hexatomic molecule." More generally, let ^ be the number of
solutions of the equation in positive integers 2x + 3^ = m, then ft arbitrary
parameters will enter into the most general representation of a covariant to a
cubic of the order m in the coefficients and the degree m in the variables. Its
graph will be a simple polygon of m sides and this will be capable of being
decomposed, in ^i essentially distinct ways, into elementary graphs consisting
either, of binary groups or, ternary groups exclusively or, the two sorts of
groups intermixed.
It may be easily shown (see Appendix 3) that every covariant of a binary
form multiplied by a suitable power of its primitive, is capable of being
represented by a rational integer function of covariants consisting, in addition
to the primitive, of covariants exclusively of the second and third orders in
the coefficients. I have already given an example of the mode in which a
graph may be augmented by an additional atom corresponding to the
multiplication of a covariant by the primitive.
The important proposition above referred to (given in Clebsch's Binaren
Formen) amounts then to affirming that any homogeneous graph augmented
by a suitable number of atoms of the same, may be decomposed, in one or
more ways, into bilooped dyads and single-sided triangles. Such a proposi-
tion ought to admit of graphical proof. The theorem has considerable
graphical importance because it enables us, in some cases at least, to discrim-
inate the true from the spurious graphs, or as we might say, p^udographs,
representing a given type. Thus, it serves to show that Fig. 14 and not
Fig. 15 is the graph to the discriminant of a cubic ; for, in accordance with
Clebsch's theorem, this discriminant, viz :
^^2 ^ 4^^ ^ 4^j3 _ 3j2^ _ 6^j^^ ^
Id
74 Sylvester, On an Application of the New Atomic Theory^ &c,
multiplied by <r becomes equal to the square of a^d — 3aJ^ + 2fr\ together
with four times the cube of {ac — ft^)% and consequently its graph, after
t.vmbination with two additional points, should be decomposable, at will, into
3 double looped lines, or into 2 single-lined triangles, which is the case with
Y\z. 14, inasmuch as its combination with two points gives rise to a simple
hexagon, but not with the second.
If we call the apices of the two figures, 14, 15, «, ft, c, rf, the true graph
i-n substituting negative signs for bonds and prefixing a sign of summation)
reads as
2 {a — by{c — (lY{a — c){b — d),
which is the cubinvariant of the quartic whose roots are a, b, c, rf, so that a
graph to an invariant of the type [3, 4: 0] gives the algebraical expression
in terms of the roots of an invariant of the reciprocal type [4, 3:0]. On
the other hand, the pseudograph treated in the same way reads as
^ (a — b)(b — c){c — d){d — a){a — c){b — d) ,
ih^ value of which is zero; a similar remark may probably be found to be
Vrif: of reciprocal graphs of invariants in general. This is abundantly con-
tnu^l by subsequent investigation ; see remarks at end of Appendix 1.
So again, if we take the graph of Fig. 42, which represents an invariant
Vj the type [3, 2 ; 1,2: 0], it reads off into
V ^B, - B,)\B, - II,) {B, - H,) ,
fc^.-longing to the reciprocal type [2,3; 2, 1 : 0], and the 2 is in fact the
discriminant of one binary quadratic multiplied by the connective between it
and another.
So if we take the graph represented in (a). Fig. 43,
^{0,-0,){0,-H){0,-K)
will represent an invariant to the type [2, 2 ; 1, 1 ; 1, 1 : 0]. If, however, we
were U) substitute 11^ , H2 in lieu of H and A", so as to form the hydroxyl
graph of Fig. 43, {b\ it would not be true that 2 {0, — 0^) {0, — H,) {0^ — H^
would represent an invariant to the type [2, 2 ; 2, 1 : 0] ; on the contrary it
would be zei*o. But hydroxyl is not an iiwariant, for to the combination of a
quadratic and a linear form there appertains no invariant of the second
degree in the coefficients of each of them. This may be easily proved by the
rule I have given at the commencement of this paper. I have gone through
tlji>5 calculation for the benefit of those new to the subject and to show how
Sylvester, On an Application of the New Atomic Tlieory^ &c. 75
the arithmetical " rule of multiplicity" is to be applied. Had I been writing
solely for algebraists it would have been unnecessary to prove so familiar a
fact. We have here i = 2, j = 2, f = 1, / = 2,w= ^I±-^' = 3. To find
(w : i,j ; i', f) we have to count the combinations
2.1
2.0
1.1
1.0
0.0
0.1
0.1
1.1;
the number of these is 4. Again to find (w—1: i,j] i'] j') we have to count
the combinations 2.0 0.0
1,1 0.0
1.0 0.1
0.0 1.1,
of which the number is also 4. Hence
A (3: 2, 2; 1,2) =4 — 4=0.
So that hydroxyl, being of the type [3: 2, 2; 1, 2], cannot be an invariant.
So far then the supposed law is safe ; but I think I see other diflSculties
in the way of its application to heteronymous types, so that if it shall be
capable of being made universally applicable, other parts of the graphical
theory, as it has been laid down, will possibly require reconsideration. What
I advance is to be regarded not as dogmatic but as tentative and ©pen to
correction.
It is obvious that not every chemico-graph, potential or even actual,
corresponds to an invariantive derivative. Of this I have already given
examples. Were the caSe otherwise we should have surprised the secret of
nature, for, as we know how to obtain all possible fundamental forms to
binary quantics, we should know a priori all possible compound radicals. As
a matter of fact the cases of algebraical invariance in nature seem to be rare
and rather the exception than the rule. Thus while muriatic acid, {H — CI),
is an invariant, self- saturating hydrogen, {H — H), is a non-invariant, there
being a linear invariant to two linear forms but not to a single one. In like
manner ozone (Fig. 16) is also non-invariantive, there being no cubic inva-
riant to a quadratic form. But there is an essential difference to be observed
between the two cases. A graph consisting of a single or an odd number of
70 Sylvester, On an Application of the New Atomic Theory^ &c.
bonds between two atoms of the same kind can never^ for any species of such
atoms, be invariantive, because no covariant of the second order in the coeffi-
cients can have an odd weight. If that were possible, then, by the theorem
of reciprocity, a quadratic function could have an invariant or covariant of
an odd weight, which is, of course, not true. Whereas a triangle of n-ads,
although not picturing an invariant when n = 2, does do so when n = 3 or
any higher number. When a homonymous graph is given in weight (the
number of bonds) and in order (the number of atoms) two of the elements of
its type {w: i^j) say w,j are known and the third i is left indeterminate. For
all values of / which make A {w: i^j) greater than zero, there will be one or
a pluralty of such graphs according to the value of A. If no value of i makes
A greater than zero, there will be no such graph possible, but it is not neces-
ASLTV to ascertain this to make an indefinite number of trials, for it is obvious
that for all values of i equal to or greater than w, A has the same value viz.
A iw: X, J), since the condition that a number w shall not be made up of
nmbers greater than i, when / is equal to w, becomes nugatory.
It will be instructive to consider the case of w = 5, j = 3, and conse-
f|uently the free valence 6 = 3i — 10; this implies that i must be at least
e«{ual to 4. But if we take i = 4, e = 2, as there is no covariant to a binary
quartic whose order is 3 and degree 2, we may be sure that A (6 : 4, 2) = 0.
Hence we have only to consider the case of i =z w = 5, e = 5. A (5 : 5, 3)
i.^ the number of covariants of the fifth order and fifth degree to a cubic of
which there is but on^, formed by the multiplication together of the Hessian
and skew-covariant. If now we proceed to form the graph corresponding to
the type [6 : 5, 3], we have the choice of two figures, 17, 18. In the former
figure there are three degrees of vacancy from saturation at A and one at each
of the points J9, C. In the latter, one at A and two at each of the points B
and C. The graph, we must recollect, is to correspond to a cubic covariant of
the fifth degree to a fifthic which is unique and indecomposable. This enables
IW to fix upon the true representation. It cannot be the graph of Fig. 17, for
that may be considered as generated by the combination of one isolated
nitrogen atom with two atoms of nitrogen, jB, C, connected by five bonds ; two
of these being subsequently welded together and bent out into the angle
having A at its vertex, [The hypothetical nitrogen pair exists in chemistry
but not as an algebraical invariant] Hence the true figure can but be that
given in Fig. 18, where the free valence is separated into the parcels 2, 1, 2, and
Sylvester, On an Application of the Nexi) Atomic Tlieory^ &c, 77
not as in Fig. 17 into the parcels 1, 3, 1. And it should be observed that, for
all higher values of i beyond 5, this will continue to be the one and only true
graph to the corresponding covariant. It thus appears that every given
homogeneous graph has an intrinsic character of capability or incapability of
respondence to algebraical in- or co-variance, irrespective of the particular
valence assigned to its atoms, and it is natural to suppose that there must
be some immediate intrinsic criterion for determining this character, so as to
dispense with the necessity of any algebraical considerations to establish it ;
but if such criterion exists, I have not yet been able to make out what it is.*
In common with this view we may consider the theory of reciprocity of alge-
braical derived forms. It has already been stated that to every mad of n-ad
atoms having a given number of bonds corresponds an n-ad of m-ad atoms
with the same number of bonds. As for example, to a quasi carbon-ad (so to
say) of sulphur will correspond a quasi sulphur- ad of carbon, the number
of bonds and consequently the amount of free atomicity remaining the same
in the two molecules. This suggests the possibility of there being some mode
of passing from a graph to its reciprocal (this reciprocity being seemingly
of quite a different kind from that which connects correlated girders or frame-
works in graphical statics). I offer the subjoined instance of such trans-
formation tentatively and with a view to stimulate inquiry, rather than as
possessing any assurance of the validity of the process employed.
Suppose the case of i = 4, j = 2, w = 4 ; the one and only corresponding
graph will be a system of 4 bonds connecting tw^o atoms A^ B. If now we
take a pair of these bonds, stretch them out, weld them together and form a
knot between them at C, and in like manner convert the other pair of bonds
into a pair knotted at i>, we shall have a graph consisting of a simple quadri-
lateral which will correspond to the case of i = 2, j = 4.
Again, suppose i = 6, j = 4, w = 12. We may consider either of the
graphs quasi in Figures 19, 20. In the first of these figures we may take
four bonds connecting respectively AC, CA^ AD, DB, stretch and weld them
together, and form a knot between them at a new point E which will then be
attached by four bonds to the atom ABCD, I mean that we may stretch out
AC, CB, to meet in E (Fig. 21) and have EC common, and in like manner
stretch out AD, DB to E and have ED common and then knot together the
*The law of reciprocity, however, exemplified in p. 74 can obviou?ly be made to supply the criterion in
question.
20
7S Sylvester, On an Application of the Netv Atomic Theory, d-c.
four bonds of the strings at E. In like manner we may form another knot
F with bonds through AB^ BC^ AD, I)C\ and shall thus obtain the reciprocal
graph of Fig. 21, where now i = 4, J = 6, iv = 12. So again it will be found
that we may distort Fig. 20 (if I can trust to my recollection of the result
•>f previous work) in two different ways into a reciprocal graph.
At the risk of provoking the ire or ridicule of my chemical friends
and the chemical public, I will venture to throw out a few remarks on the
substructure, so to say, of the accepted theory of atomicity and to offer a
sugirestion as to a possible mode of getting rid of some imperfections under
which it appears at present to labor. First there is the inconsistency of
admitting the isolated existence of single atoms of mercury, cadmium and
zinc, as monads with their bonds or tails absorbed or suppressed or else
^winging loose and unsatisfied in direct opposition (as it seems to me) to the
fundamental postulate of the theory. Next, one cannot get over a somewhat
uncc»mfortable feeling at the representation of isolated oxygen in the state of
• •»>ne by a triangular graph, which, although conceivable, is supported by
no analogous case unless that of baric peroxide, or any similar graph, be
regarded as such. Thirdly, there is the vague and unsatisfactory (not to say
unthinkable) explanation of the variability of the valence of a given atom
bv what Dr. Frankland calls *' the very simple and obvious assumption that
one or more pairs of bonds belonging to the atom of an element can unite and
having saturated each other become, as it were, latent."
Now these stumbling blocks to the acceptance of the theory may be
removed by one simple, clear and unifying hypothesis, which will in no wise
interfere with any actually existing chemical constructions. It is this: leaving
undisturbed the univalent atoms, let every other n-valent atom be regarded
as constituted of an n-ad of trivalent atomicules arranged along the apices of a
polygon of n sides. Thus, sextivalent, quinquivalent and quadrivalent atoms
in their state of maximum valence will be represented by Figures 22, 23, 24,
where the letters denote trivalent atoniicvles. When the valence is reduced by
two we need only conceive any one of the side loops doubled or a new loop as
formed by the coalescence of a pair of free bonds or tails, and when in the
Figures 22 and 23 the valence is reduced by 4, we may in like manner either
suppose existing loops doubled, or fresh ones inserted, or both changes to go
on simultaneously, by the coalescence of two pairs of tails. We have thus a
c^^n<^5eivable and conformable-to-analogy method of accounting for the varia-
Sylvester, On an Application of the New A tomie Theory, &c. 79
bility in question. So likewise, a trivalent atom with maximum state of
valence will be represented by Fig. 25, and when univalent by Fig. 26.
Again, an isolated zinc element will have for its graph Fig. 1, (6), the two
letters Z signifying the zinc atomicules, and so in like manner isolated cad-
mium and mercury may be represented. On the other hand 0^^ isolated
oxygen in its ordinary state, will be represented by the graph of Fig. 27,
whilst ozone will have for its representative graph the well known Kekulean
hexad (which, in its importance to chemistry, would seem to vie with Pascal's
mystic hexagons to geometry) represented in Fig. 28, where as in Fig. 27,
each letter represents an atomicule of oxygen. So an isolated element of
carbon would be represented by the graph of Fig. 29.
This hypothesis of atomicules, if unobjectionable on other grounds,
would not be open to the charge of having any tendency to disturb or com-
plicate the existing graphology ; for we should still be at perfect liberty to
substitute for the graphs {a) of Figures 30, 31, 32 the abridged notation (5),
and should naturally do so when considering the relations of atoms to each
other. The beautiful theory of atomicity has its home in the attractive but
somewhat misty border land lying between fancy and reality and cannot, I
think, suflfer from any not absolutely irrational guess which may assist the
chemical enquirer to rise to a higher level of contemplation of the possibilities
of his subject. I have therefore ventured to make the above suggestion.
Chemical graphs, at all events, for the present are to be regarded as
mere translations into geometrical forms of trains of priorities and sequences
having their proper habitat in the sphere of order and existing quite outside
the world of space. Were it otherwise, we might indulge in some specula-
tions as to the directions of the lines of emission or influence or radiation or
whatever else the bonds might then be supposed to represent as dependent
on the manner of the atoms entering into combination to form chemical sub-
stances. Such not being the case, what follows is to be considered as having
relation to mere algebraical atoms, or atomicules (quantics) and their bonds
which may be regarded as represented by the linear factors of such quantics.
Let us consider a symmetrical trivalent atomicule whose three bonds or
rays make angles of 120° with each other. Calling r, t', t^, the tangents of
the angles which the axis of x makes with its rays, we have
T + \/"3 ^, T — \/^
1 — V3T 1 + -v/3t'
80 Sylvester, On an AppUcation of the New Atomic Theory^ &€.
so that its equation will be easily found to be
(1 — aT2)r^+(9T — 3T^)a:2y+(9T2 — 3)a:/+(T« — 3t)/ = 0,
which may be identified with the standard form
a^ + ^hx^y + Zcxf + rf/ =
by writing a = l — 3t^ = — c, 5 = 3t — t^ = — d.
Suppose the three atomicules to become condensed into a single atom after the
manner of the graph of Fig. 25. The combination will be represented by the
cubic CO variant (see Tables des Invariants et Co variants, Table V, annexed to
Faa de Bruno's *' Theorie des Formes Binaires ")
{a4 — Sabc + 2b') x" + {3abd — bac" — 3b^c) x'y
+ {Sbc" + 6b''d — 3acd) xf + i^bcd — ad^ + 2c') f ,
which, for the present cage, becomes
2 (1 + t^)'[(3t — T^) x" + (9t^— 3) x^y + (ar^— &r) a:/ + (1 — ar^) /].
Hence the new ray-directions will have for their equation
— da^ + Scx'y— 3bxf + af = 0,
or the pencil of the atom will be identical with that of each of the separate
atomicules, but accompanied with a rotation (whatever that may mean) of the
whole pencil of rays through a right angle in its own plane. Again, suppose
that only two atomicules are brought into connexion as in (a) of Fig. 30. The
quadricovariant which expresses the atom (Faa de Bruno ante) is
{ac — b^) x^ + i^d — be) xy -{- {bd — c^)
which here becomes — (1 + t^)*^ {x^ + y^).-
Hence the ray-directions will be given by the equation
f + x' = 0, y = ziz^ V^a ,
which we may, if we please, according to the usual convention concerning the
square root of minus unity, explain by supposing that the original rays are
situated in planes perpendicular to the joining line XX, and that these are
replaced by two rays lying in opposite directions along the line XX, where
the atomicules are condensed into one atom. But it would be idle to pursue
this speculation further.
The most remarkable point in the theory which I have endeavored to
unfold in the preceding pages is the relation between it and that of reciprocal
types.
Sylvester, On an Application of tlie New Atomic Theory^ &c, 81
We have seen that the graph to an invariant of one type read off as it
stands (each bond being construed as the sign minus) with the sign 2 pre-
fixed expresses an invariant of the reciprocal type.
This rule may be extended from homogeneous to heterogeneous graphs,
provided only that the reciprocity be totals by which I mean that every i and
every j in the type [i^j; i'^j';i\f, : 0] are interchanged. It may be
observed, in passing, that in the case of types to which resultants belong, the
type is identical in form with its total reciprocal. As ex. gr. boric anhydride
(consisting of two of boron and three of oxygen) is of the type [3, 2; 2, 3: 0].
On referring to " System of Cubic and Quadratic," Salmon's Lessons,
third edition, p. 179, it will be seen that besides the resultant there is another
invariant represented in Dr. Salmon's notation by *'A (0, 2) X ^ (2, 1)"; a
linear combination of these two with arbitrary multipliers will express the
most general form belonging to the type in question.
From, the property of these types being their own complete reciprocals,
it follows that a complete set of independent graphs of any such type will
represent the constitution of a complete set of independent forms belonging to
the type. Thus, in the case suggested by boric anhydride we have the two
independent graphs of Figures 33, 34. Hence the complete representation of
the invariants appertaining to the self-reciprocal diadelphic type [3, 2; 2, 3: 0]
is X ?7 + |t^ F, where U is the resultant {a — a) («—/?) {a — y) {b — a) {b — ^) {b — y)
and Fis 2 {a — y){a — (3){b — a){b — y){b — a){(3 — a). ?7is derived from the
graph of Fig. 33 by replacing the several (9'*s by a, (3, y, and the B'b by a, ft,
and Fin like manner from the graph of Fig. 34.* This latter graph is replace-
able by the disjoined graph of Fig. 35, to which, by the rule for combination
of graphs, it is easily seen to be equivalent.
Hence, instead of X?7 + i^^F we may write XV -{- [iV' where F'
= 2 (a — /3)^(a — by {a — y){b — y) ; a, b of course will be understood to be
the roots of a general quadric and a, /3, y of a general cubic. A very good
similar instance of this kind of equivalence is afforded by the quadrinvariant
of a quartic whose type is [4, 2:0]. The reciprocal of this, viz. [2, 4: 0],
may be represented, either by the connected graph of Fig. 36, or by the
disjoined one of Fig. 37, and accordingly the noted quadrinvariant ae — 4W
+ Sc^ may be expressed (to a numerical factor pres) either by the sym-
*In this figure on the side opposite to BB, a third letter has been accidently omitted.
21
82 Sylvester, On an Application of the New Atomic Theory^ &€.
metricAl function S {a — c){a — d) [b — c){b — d) corresponding to the first,
or by 2 {a — by^{c — dy corresponding to the second graph. Again, let us
consider the contrary types, [4, 3: 0], [3, 4: 0]. The former has for its
graph Fig. 38, and admits of no other representation. This gives
for the discriminant of the cubic which belongs to the contrary type. The
latter may be figured chemically by the graph (consisting of two molecules of
boron) of Fig. 39, or by the equivalent Fig. 27 (capable of being derived from
it by the mechanical rule for conversion of graphs). These two latter, alge-
braically speaking, will be pseudographs, because 2 (a — ^Yiy — ^Y and
2 (a — /3) (/3 — y){y — h) [h — a) (a — y) {(i — 5) are each zero. The graph of
Fig. 27 may be mechanically converted, in the manner shown in the preceding
case, into the graph of Fig. 40 ; but the type of the colligation remains
unaltered by this conversion and whichever of the two we employ, we obtain
s (a - mr - «)^(« - y) (/? - 5)
as the representation in terms of the roots, of the cubic invariant to the
quartic, viz. to a numerical factor pres
ace — b^e — ad^ -f 2bcd — c^ .
Thus we see that the graphical method suggested by the theory of atom-
icity is a real instrument not merely for the representation but also for the
calculation and comparison of algebraical results. The important bearing
upon it of the principle of contrary or reciprocal graphs, renders it desirable
that I should put the algebraical theory or law of reciprocity, in its most
complete form, before my readers ; it will form the subject of Appendix 2.
I might have noticed explicitly at the commencement of this paper,
instead of tacitly assuming it as I have done, that the chemical fact of a com-
pound molecule playing the part of an atom with a valence equal to the free
valence of the radical, is the precise homologue to the algebraical fact that
every invariant or covariant of a covariant, or set of covariants, to a quantic,
or system of quantics, is itself an invariant or covariant to such quantic, or
system of quantics ; and again that Regnault's chemical principle of substi-
tution and the algebraical one of emanation* are identical ; and again, the
* By which I mean in this place the operation upon an invariant or covariant of the symbol (a'<J,+ ^^\-\-- • •)
performed any number of times in succession ; a, b, for instance, may refer to Hydrogen (ax + by) and a', b^ to
Chlorine {a\x + b^y)t and then the emanantive operator, according to a notation used, if I mistake not, by
Professor Clerk Maxwell in his theory of poles, might be denoted by CI 6^ .
(CL)
Tig.Z
O
Tig.S
(h)
" A
"^ II II
II II
f» o o
II
o
^t^jt
■Fig' 8
A
Fig.ia
Fig. 9
Fiy.dU
O j»
'.^S
A.
C 2»
Fig.d^ Fiff.20
C 2>
Fiy. Uk Fitf.2S
X X
Fig. 22
Fig. 23
Fif.28
o o
Fiff.Sy
c a
Fig. 30
(at '(b) '
Fiff.32
(a) (b)
Fig. Sif
Kl
o o
A
\B O
Fig.St Fiff.38 Fig.SS J'ig.iO
o O C S B O o'
o o
c s s
n
o
Jtr
Fig.M^ Fig. 4t2
Tig. * 4
^ X ^ ^' M
F'ig. 4tS
o o
(b)
J£
84 Sylvester, On an Application of the liew Atmnic Theory^ &c.
the way of its reception, a very lion in my path, so formidable that, for a
time, I thought that it would be necessary, either to abandon this law, or else
to admit the unwelcome conclusion that not every type of invariant was
susceptible of graphical representation."
But further consideration has shown me that this apprehension was
entirely groundless owing to an algebraical fact on which I had not previ-
ously reflected, but which this difficulty forced upon my notice. The difficulty
in question arose out of the expressions given by M. Hermite and le pere Jou-
bert respectively for the skew invariants of the binary quintic and sextic. I
shall first address myself to the consideration of the former. Following Dr.
Salmon's notation (Lessons, 3d Edition, p. 230), let a,/3, y, 5, e be the roots of a
quintic, and let
F={a-^){a-e){h-Y) + {a-Y){a-hW-e)
G={a-^){a-Y){e-h) + {a-h){a-e){(i-Yy
11 = {a-li){a-h){e-r) + {a-y){a-B){h-^) .
Then it will be found as will presently be shown that the product F,G,H is a
symmetrical function of the four roots /?, y, 5, e, consequently, on forming
four other similar products symmetrical in respect to a, y, 5, e : a, /3, 5, b : a,
/3, y, e : a, /?, y, i respectively, the product of these five products will be sym-
metrical in respect to a, ^, y, 5, e and being a function of the differences of the
roots of order 18 and of weight 45, *?'. e. of the type [45: 5, 18], must be
(paying no attention to a mere numerical factor) /, the skew invariant to the
quintic.
Now consider the type reciprocal to this, [45 : 18, 5], (monadelphic like
the preceding), and expressing the invariant of the fifth order to an octode-
cadic. Suppose this has a graph. It will follow from the law of reciprocal
graphs that / may be expressed under the form
X{a-^)'{a-rY{a-Sy{a-ey{j3-yy{^-6y{^-eyir-Sny-sY{8-e)\
where a + b -\-c+ .... -zz 45 and each letter a, /?, y, 5, e is conditioned to
appear the same number of times, which at first might seem contradictory to
what has just been established, but in reality is in perfect accordi^nce with it.
For imagine the product of the 15 ([naniitiesFGHF&ffF'&'H''F''G''Il''F^G'^H'''
to be actually written out giving rise to 2^^, or 32768 terms, and to each of
these terms prefix the sign 2 indicating that the sum is to be taken of the 120
values which it assumes on permuting the 5 letters a, /?, y, 5, a. The sum of
Sylvester, On an Application of the New Atomic Theory^ &c, 85
all these partial sums is 120/; hence some, at least, of them cannot vanish.
Let 2Tbe any one that does not vanish. Then 2T is a function of the differ-
ences of the roots of the same weight and order as the entire expression ; it is
therefore to a numerical factor pres identical with /, just as every fragment of
a mirror is itself a mirror, or as every particle of diamond dust, a diamond.
Thus, as many distinct non-vanishing forms as there may be of 2T, so
many different graphs to the quint-invariant of a binary octodecadic shall we
be able to construct agreeing respectively with the different representations of
/ of the form
2(a — /?)«(a — y)^(a — «)•
and it is probable that the virtual equivalence of all these several graphs may
admit of being made out by inspection, as we saw was the case with the
two graphs (one dissociated, the other connected) corresponding to the two
algebraical representatives of the quadrinvariant of a quartic. Thus, what
seemed, at first sight, to be fatal to the admissibility of the algebraico- graph-
ical theory only serves to set in a clearer light its value as an instrument
of research.
If we analyse M. Hermite's form of the skew invariant* *to the quintic
we shall see that it depends upon this simple but not obvious fact, that writing
F = {c, d) {a— b) + {a, b) {c — d)
G={b, d) [a—c) 4- [a, c) (d — b)
11= (6, c) {a — d)-\- {a, d) \b — c)
and interpreting any such quantity as («, b) to mean cither 1 or {a + b) or ab
the product FQIIis a symmetrical function of «, 5, c, d, because on interchang-
ing any two letters (say ex. gr. c, d) that one of the three quantities F^ G, H (in
this example H) in which those two letters are affected with the same sign,
will remain unaltered in value whilst the other two (here G and F) change,
each into the negative of the other.
Consequently we may interpret («, b) to mean {e — a) {e — b) and then
the product of the five products corresponding to FGH is a function of the
*I am wont to compare in my mind this symmetrical and translucent form to the Pitt Diamond »nd Pdre
Joubert's to the Koh-l-Noor. In Note D to Appendix 2 a method Is given whereby these forms may be trans-
muted into one another subject, however, to the bare possibility that the one, put into the algebraical alembic
at a certain stage of the process, instead of passing into the other may, so to say, evaporate and bo reduced to
nothing. In the theory of forms, all-embracing Zero is the source and reconciler of contradictions, because,
algebraically speaking, everything is contained in nothing ^ njid so in a morphological sense ** nought is every-
thing *' though not " everything is nought."
22
86 Sylvester, On an Application of the New Atomic Theory^ &c.
coeflScients which expressed in terms of the differences of the roots will be of
the weight 15 and of the order 1 • 6 + 4 • 3 or 18 because in one of the 5
products each letter will enter in six dimensions and in each of the other 4
products in 3 dimensions ; thus in FGH^ ^ will appear, but in each of the
other 4 products ^ will be the highest power of e. Hence the quindenary
product is the invariant in question. No further step is necessary, the proof
is complete as stated.
This remark will enable us to illustrate the process of transformation,
which I have compared with grinding a diamond into dust, by an example
that can be completely pursued to the end. For let us now regard a, ft, c, d as
the roots of a binary quartic ; then (a — h + c — r/) (a — c + d — ft) (a — d +
ft — c) will be a diiferentiant thereto of weight 3 and order 3 ; it will, in fact,
represent the root-diflferentiant of the skew sextic covariant.
Imagine this multiplied out without disturbing the marks of coupling so
as to give 8 terms or fragments analogous to the 32768 fragments spoken of
in the preceding case. These terms will be of only four different patterns,
one of the pattern {a — ft) (« — c) {a — d) , three of the pattern {a — h)a — c) (ft — c) ,
three of the pattern (a — ft) (ft — c) {d — ft) and one of the pattern {c — d){d — ft)
(ft — (?). Prefixing 2 to each of these pattern terms to signify the sum
resulting from the 24 permutations of a, ft, c, rf, we know a priori that not all
of these can be zero since a linear function of them will be 24 times the differ-
entiant in question, and on examination we find that the second and fourth 2
will vanish, but that the first and third will not. Accordingly, we shall have
two new expressions 2 (« — ft) (a — (?) (ft — ^), 2 (« — ft) (ft — c)Q> — rf), each of
which represents a differentiant of the same type as the original one, and this
type being monadelphic or enparametric, the original product and these two
sums will only be different representations of the same differentiant. Thus
we see that each independent form belonging to a given type is susceptible
(when expressed as a function of the differences of the roots) of a number
of distinct phases, or, as we may express it, an algebraical form, in this
theory, is in general polyphasic and accordingly its Icon or linkage expo-
nent will be in general polygraphic, and each phase will have its own appro-
priate graph. It is a work of some difficulty, in general, to recognize the
substantial identity of the different phases of the same algebraical form, and
in like manner it may not, in all cases, be easy to recognize the substantial
identity of the different graphs of its Icon, but sufficient has been shown to
Sylvester, On an Application of the New Atomic Theory^ &c. 87
m
indicate the possibility and method of establishing such identity. The more
I study Dr. Frankland's wonderfully beautiful little treatise the more deeply
I become impressed with the harmony or homology (I might call it, rather
than analogy) which exists between the chemical and algebraical theories.
In travelling my eye up and down the illustrated pages of " the Notes," I feel
as Aladdin might have done in walking in the garden where every tree
was laden with precious stones, or as Caspar Hauser when first brought out
of his dark cellar to contemplate the glittering heavens on a starry night.
There is an untold treasure of hoarded algebraical wealth potentially con-
tained in the results achieved by the patient and long continued labor of our
unconscious and unsuspected chemical fellow- workers.
We have seen that M. Herraite's beautiful expressions for the skew
invariant of the quintic proves its own character. A similar analysis may
be applied to pere Joubert's equally beautiful and even more remarkable
expression for that of the sextic. M. de Bruno's statement of this. Table
IV*®, contains two very perplexing typographical errors, viz. 4th line from foot
of page, in F©, x^x^ {x^ + ^^o — ^z — ^-2) should read X1X2 {x^ + ^0 — ^s — -^Oj
and 3d line from foot of page, in Wq, x^x^^ {X2 + x^ — Xoo — •t'o) should be
Xa^Ti {xx + 0^3 — x.y^ — j*o) • Moreover, the form in which the expression is
presented in M. de Bruno's pages tends to mask its true nature and to suggest
an analogy, which has no existence in fact, between it and M. Hermite's form ;
the latter is intrinsically a quinary group of triadic products, but such repre-
sentation in the case of M. Joubert's form is purely conventional and confus-
ing, it really being a single indecomposable quindenary product. Call «, 5, t?,
dy e^f the six roots of a sextic, and let ab] cd\ ef be any one of the 15 duudic
synthemes^ which can be formed with them, and
ab.c -{- d — e — /
F= -^^J\^cd. e + f—a — b
+ ef. a + b — c — d
The external sign is arbitrary, but must be considered as determined once for
all for each of the 15 values of F. The product of these 15 values is a sym-
metrical function of the roots. For suppose any two letters, as a, &, to be
* A duadic synthcme of 2n letters is a combination of n duads containing between them all the letters. In
it the order of the duads and of the letters in each duad is disregarded. Uence the number of such is
or 1'8'6 * ■ * (2n — 1). For an odd number of letters simple synthemes do not exist but in lieu of them we
may construct diplo-synthemcs containing every letter taken twice over.
88 Sylvester, On an Application of the New Ati/inic Theory^ &c.
interchanged ; then three of the factors F in which a and h are coupled will
undergo no change, but the remaining twelve will evidently be resoluble into
six pairs reciprocally related, so that each F of a pair is transformed either
into the other or into its negative and on either supposition the product of the
pair remains unaltered in value. Also this product is a differentiant, for XK
operating on any one factor evidently reduces it to zero. It is also of the
weight 45 and of the order 15. Hence the product of the fifteen values of F
is the skew invariant to the sextic.
It seems desirable to make the differentianiive character of the form self-
apparent. This may be done by virtue of the remark that i F J^ay be
replaced by the form
(a_rf) (ft_/) ^c-e) + {a-f) {b-d) (c-e)
+ (a _ c) (ft _ e) (d-f) +{a-e){b- c) {d -f)
+ {<i-c) (b-f) {d-e) + {a-f) (b-c) (d-e)
+ i^-d) (b-e) (c-^f) + (a-e) (d-b) (c-f)
This sum contains 6 1 terms, of which 48 are the terms in F taken 4 times
over, and the other 16 are the 8 quantities ace^ hdf acf bde^ bce^ adf bcf\ ade,
each appearing twice with opposite signs. If we expand the product of the
15 values of F, we shall obtain 35,184392,568832, or upwards of 35 billions of
terms distributable among a certain number of patterns ; on prefixing 2 to
one of each pattern a certain number of such sums will be zero, but the
remaining ones of which there must be some (and there will probably be a very
large number) will all be (except as to a numerical multiplier) identical with
each other and with pere Joubert's formula. We see by these examples that
there is a sort of polymorphism or pheno-polymorphism, as it may be termed,
which is of a much more superficial character than and ought to be carefully
distinguished from true polymorphism, eteo-polymorphism as we may call it,
and this distinction as it has a marked bearing upon the theory of algebraical
linkages, it is reasonable to expect may not be without importance in the
study and construction of chemical graphs. Although I have been dealing,
in what precedes, with particular cases, the reasoning is general in its nature
and leads to conclusions which I will proceed to express in exact terms.
Let us understand by a permutation-sum of a function of letters belonging
to one or more sets (n, n\ w", being the number of letters in the respec-
tive sets) the sum of the HnHnTln'' values which the function assumes
when the letters in each several set are permuted inter se; and let us under-
Sylvester, On an Apidication of tlie New Atomic Tlieory^ <S:c\ 89
stand by a monomial d liferent iant one which (with the usual convention as to
a=.l) may be expressed as a permutation -sum of ^ single product of differ-
ences of roots of the parent quantic, or quantic system ; then in the first
place it has virtually been proved, in what precedes, and is undoubtedly true
that every raonadelphic differentiant is monomial, and it may easily be proved
in like manner that a differentiant of multiplicity k may be represented by
the sum of k monomiiil differentiants.
For greater simplicity let us confine ourselves to the case of monadelphic
invariants and let us consider any two such belonging to reciprocal types;
then the algebraical value of either one, in terms of the roots of its parent
quantic or quantic system, will be represented by the permutation-sum of the
product of the differences of every two letters in the other taken a^ many
times as there are connecting bonds between them, such letters being for this
purpose regarded as the roots in question. Hence also we may derive the rule
previously given for determining whether or not any given graph, in which the
number of bonds is equal to half the toti-valence, represents or not an alge-
braical invariant — the condition of its doing so being that the permutation-
sum of the product of the differences between the connected letters (each
bond giving one such difference) shall be other than zero. This rule will
stand good whether the type of the graph be monadelphic or not.
A very simple instance occurs to me of the monomial law for mona-
delphic types. Let a, /?, y be the roots of a cubic. It will easily be found that
the type (4 : 3, 4) to which
((a - ^y + (a - yY + {3 - yy^-
belongs ia monadelphic ; prefix to it the sign of summation, which is merely
equivalent to multiplying it by 6. It will not be a monomial permutation-
sum as it stands, but it may be replaced by 22 (a — iSy {a — yY or 2(a — i3y
each of which monomial sums is a half of
((a - l^r + (a - yY + {[3 - yY)'-
Postscript. Subsequently to the printing of the foregoing sheets I have seen in an editorial
notice in the English Journal Nature (March 14, 1878) a statement of the claims of Dr. Frankiand
to be the discoverer and first promulgator of the law of atomicity, and I appear unconsciously to
have done injustice to this great English chemist by attributing the discovery to Kekule. I derived
my impression on the subject from the popular belief and from the account of it given by Wurz in
his " Histoire des doctrines ch^miqnes." If the facts of the case are as set forth in Nature and
admit of no qualifying statements, I am unable to understand how such a discovery as that of va-
lence or atomicity, which furnishes the master-key to our knowledge of the transformations of niattcr
and raises chemistry to the rank of a mathematical and predictive science (it was previously only
arithmetical), can have escaped receiving the award of a Copley Medal from the society in whose
«3
90 Sylvestek, On Ilermifes Law of Reciprocity.
Transactions it appeared. I can hardly imagine that, if the first annonncemcnt and proof of ani-
versal gravitation or the circulation of the blood had been commanicated to the world in a paper
inserted in the Philosophical Transactions in these days, its author would ha?e failed to receive for it
the highest mark of recognition in the power of the Royal Society of London to bestow, and in my
humble judgment the law of atomicity in its far-reaching importance and the labor, and mental
acumen required for its discovery, stands fully on a level with either of these great landmarks in the
history of natural science. It seems also from the same article in Nature that my distinguished friend,
Professor Crnm Brown, to whose personal teaching at Edinburgh I owe the very slight acquaintance
with the subject I can lay claim to, was the first to use the admirable method of chemico-graphs.
The conception of hydro-carbon graphs as " trees with nodes, branches and terminals " and the
indispensable notion of constructing them by starting from "an intrinsic central node or pair of nodes,
so as to get rid of the otherwise unsurmountable difficulty of having to recognize equivalent forms
appearing several times over in the same construction,'- are exclusively my own and were used by me
in ray communications with Professor Crum Brown on the subject and stated by me in a letter to
Professor Cayley, who has adopted them as the basis of his own isomerical researches. In the account
of this method given in German chemical journals I am informed that all reference (or at least all
adequate reference) to my name as the author of it " fine by degrees and beautifully less," has at
length entirely evaporated. M. Camille Jordan was led by quite a different order of considerations
and with quite a different object in view to a discovery of the same centres before me, but I was not
acquainted with this fact when I rediscovered them and made the application above mentioned.
The idea of this application stands in the same relation to Professor Cayley's perfected use of it, as
his idea of the use to be made of the equation ^ (w: i.j) -.: the number of linearly independent
covariants of the type [^i,j: ij — 2m'] statida to my completed method founded thereon, for obtaining
the scale and connecting syzigies of the irrtducible covariants to a quantic, laying me thereby under
an obligation which I should take it in very ill part if any translator of my papers on the subject*
failed to acknowledge in unmistakable terms.
The hydro-carbon graphs, it may be noticed, belong to the limiting case of chemico-graphs;
where no cyclical system of bonds connects any groups of atoms in a graph, it becomes an arbor-
escence.
I have found it a profitable exercise of the imagination, from a philosophical point of view, to
build up the conception of an infinite arborescence and to dwell on the relations of time and
causality which such a concept embodies. An example of the good to be gained by these limitless
mental constructions (new tracts and highways, so to say, opened out in the all-embracing "grand*
continunm " which we call space) is afforded by the valuable applications to the theory of local proba*
bility and the integral calculus in general made by Professor Crofton (my successor at Woolwich)
of his new idea of an infinite reticulation (warp and woof), every finite portion of which contains
an infinite number of meshes, being formed by the crossings of two sets of parallel lines all infinitely
extended in both directions and those of the same set equidistant and infinitely near to each other.
So the largest idea of an arborescence is that of an infinite number of nodes with an infinite number
of branches proceeding from each of them.
APPENDIX 2.
Note on M. Hermite's Law of Reciprocity.
I TAKE for granted that the treatise of M. Faa de Bruno represents this
theory as it at present stands, in which ease it seems to have made no advance
since it was first promulgated by M. Hermite in his well known paper in the
Sylvester, On Hermite's Law of Beciprocity. 91
Ciimbridge and Dublin Mathematical Journal, 1854. It will be seen, however,
I think from what follows, that it admits of being presented in a somewhat
simpler and more general form. It rests essentially on the proposition of
reciprocity in the theory of partitions that {w : i,j) zz {w: j, i), from which it
follows as an immediate consequence that the number of arbitrary constants
in the general covariant (or invariant) whose type is [w: i, j], is the same as
that whose type is [iv : j, i] since that number will be A(w : i,j) zz A{w: j, i)
for each. Let now <^ [a, b, c, , . , , l) he any differcntiant of the order j in the
coefficients, and of the weight w to a binary quantic F{Xy y) of the degree i in
the variables; then <^ is the root of a single covariant whose order is ^' and
degree in the variables ij — 2w. Let <^ be expressed (as from the definition of
a differentiant must necessarily bo possible) as a function of the differences of
the roots ai, 0C2, . . . . a^ of F when y is made unity. For any difference a^ — a,
substitute -— . r- • -7- ? ^^^ ^^^ ^ ^^ converted into cb by this substitu-
dXp dy^ (Ivg dyp
m
tion. . Now operate with <^ upon the product of the j forms G (ar^, yi),
G^ (-^'ii ^2) 7 • • • ^ {^i^ !/i)j Gr (.r, y) signifying the general form of the degree J in
the Variables, and after the operation has been performed turning each
subscript x into x and each subscript y into y, after the manner of Professor
Cayley's original method of generating invariants or covariants as '' Hyper-
determinants," we shall thus obtain an in- or co- variant to a form of the
degree j which will bo of the order / in the coefficients and of the degree
ij — 2w in the variables, for there are w fiictors in <^ and each factor is of the
second dimension in two of the .r'*s and the corresponding two y's. Thus we
shall have passed from a form of the type [?*, j: ij — 2w\ to another of the type
Ei» ^- ij — 2w], or which is the same thing, from one of the type \w\ i, j] to
another of the type \}o : j, i].
This latter may be called the inioffe of the first. For facility of reference,
let the number of arbitrary parameters in the one and the other type T^e
called the multiplicity. If we repeat upon this image the process by which
it was deduced from its primitive, we shall obviously get back the original
type, but it by no means follows that if the multiplicity exceed unity, we
shall get back the primitive form itself. It may be possible to revert to the
same type without reverting to the same individual specimen of it ;* and such,
we shall presently see, is what in general happens.
♦Just as, if I rightly understand the explanation given of fluorescence, a ray of light may give birth to
some other form of motion and that again to another ray of light but of a different color from the first. The
theory of reciprocity treated of in the text is, in fact, a theory of alternate generation.
92 Sylvksteb, On IIennit4^'$ Imw of Beciprocify.
Before proceeding further I shall give a very simple methodical rule for
finding the image to any given invariantive form. Since, for any given value
of /. the form and its image are each given when their root-diiFerentiants are
respectively given, it will be suflBcient to assign the law for passing from the
differentiant of the primitive to that of its image.
For this purpose, let the given in- or co-variant be expressed in terms of
symmetrical functions of the roots of the quantic when the leading coeflBcient
(a), is made equal to unity. Then it will consist of terms, any one of which,
apart from its numerical coeflScient, will be of the form
S (aitti .... a^)" {SxSi .... S^)^ ^yiYi - • - • y,^' t^i^ .... K)^ ....
a/ju . . . a^, 3i32 . . . 3^- yiYi . . . /„ etc. being all distinct and comprising
between them all the / roots and of course u + -*' + 3:i + etc will be equal
to the weight ; to pass from a differentiant expressed in terms of roots of a
given quantic to the expression in terms of coeflScients of the allied quantic of
its image it will be found that the only thing necessary is to change any such
factor as a^ (where a is any root of the given quantic) into C\. the coefficient
of the term containing y^ in the allied one. This rule is a consequence
(obtainable by ordinary algebraical processes) from the method above ex-
plained, where it is to be borne in mind that in order to obtain the image
from the given form we have only to substitute for each root a, which
rfr
occurs in <^, the fraction -^ and to multiply the result by such a power of
-— - — -- , as will just serve to make it integral. A much simpler
demonstration of this rule will be given in the sequel, and it will be shown that
it not only holds good for deriving the leading term of the reciprocal (in the
case of a covariant) from that of the primitive (i. e. the root-diflferentiant of
the one from the root-diflerentiant of the other) but that it is applicable to
deriving the whole of one expression from the whole of the other.
As an example, take the diflerentiant whose type is [3: 3, 3], the root or
base of the skew covariant to a cubic (a, 6, c, d \ .r, y)'. Its value is (rd — ^bc
+ 2h^ ; expressed in terms of the roots a, .i, y, making a = 1, this becomes
a i'v —^ i«_+J^+ r) M + «y + Sy) .-> (a + 3 + y?
: 7 Q "^ ^ 27 '
or ^l I 27a;3y — 9 (a + ^ + y) (ai + ay + 3y^ -f 2 (a + .i + yf | ,
or 1 1 22a» - 32a^^ + 12a^y } ' '• ^- ^ { 22<^/ - SSaVV + V^'3y } .
Sylvester, On Hemiite'a Law of Reciprocii^. 93
Applying the rule, tbis becomes converted into
~ I QCIC — 186;aa + 12C? J ,
or, reverting to the letters a, h, c, d, the image becomes the primitive affected!
with the factor — and may be seen to be its own conjugate. Or again, let
the primitive be the discriminant of a cubic, i. e.
i (« - m" -rm- r)' or (a'/3 + (3V + f" - "(3' - l^f - r"')' ;
this is equal to
Hence, by our rule, the image will be
^ (Qc^c^c, + 12c,eac» — Gc^(i — 66^ — 6cie,) ,
or, using a, b, c, d, e in lieu of Cg, c,, c^, Cg, c^, we obtain the form
— 07 i^^^ + 2^*^*^ — '^^^ — ^^ — *^*)>
2D . . '^ *
t. e. ^ , where Ziis the well known quad rin variant to a quartie b c
c d
Treating this quadrin variant as a function of the roots of a biquadratic form
and proceeding aa before to form its image, we shall obtain - a second image
which will be a numerical multiplier of the original invariant.
But now let us consider the case of polyadelphic forms belonging to
reciprocal types and for greater brevity, as the calculations are necessarily
long, take a quantic of the self-contrary type [w.- i, t], as ex. gr. [6: 4, 4] which
belongs to the covariant of the fourth order and fourth degree to a quartie.
This will be diadelphic; its general form is a linear combination of two pro-
ducts, one of the quartie itself by its cubinvariant, the other of the Hessian
by the quadrinvariant. It will therefore have for its leading coefficient the
differentiant
Xa {ace + 2bcd ~ ad^ ~ (f — f^e) + fi {ac — ^) (ae — 4ftrf + 3c*) ,
aay Ti-U -^ (iV. L<et us first find the image of U. Expressed in terms of the
roots a, /?, y, h, it is
1 (a/3 + ay + ai + /3y + /i5 + yh){a(iyh)
94 Sylvester, On Henaites Law of Reciprocity.
+ ^ (a + ..i + y + h) (a3 + ay + ai + 3y + ;3h + yb) (a.3y -f aSb + ayb + ^yb)
— iA {oL:3y + a;36 + ayb +3ybf — -; {aS + ay + ab-\- 3y + 3b + yby
— j^ (« + ;3 + y + bf a3yb ,
which is
[6] (g-yy) [24] («.yy) + [48] {a3fb') + [12] jaY-f) + [1 2] («..?yy)
6 "^ 48
_ [4] {a',r-f) + [12] (a,3y^y )
16
[6] (a»,3») + [90] {a3f}^') + [72] {a.yf) +. [24] (a^^r) + [24] {a^ yb')
216
_ [4] {a3yh') + [12] (o^V*!)
where any term, as ex. gr. [48] (aSy^b^), means the sum of the quantities of
the type a^^3yh each taken a sufficient number of times to make up 48 com-
binations, so that it is identical in meaning with 82 (oiiSy^b^) in the common
notation. This convention is useful in saving the unnecessary labor of per-
forming divisions in this first part of the process which have to be exactly
reversed by multiplications in the transformation process which follows. The
value of the above sum is, for purposes of transformation, equivalent to
which gives for the image of U
^ (36V + 6abcd — 4a(f — 4db' — aH^)
or ^-{U — F), where it will be observed that ( F — ?7) is identical with the
oo
discriminant to (a, b^ c^ d\ or, yY. Let us now proceed to find the image of
{U — F). Using a to denote the sum of the combinations of a, /?, y, h taken
i and i together, where a, /3, y, h are the roots of the general quartic, we have
~ 192 "^ 16 54 64 16
1
1728
(9j?(t^ + 108(Ti(T2(T3 — 32(r^ — 21a^, — 108a|).
Sylvester, On Hermite^s Law of Reciprocity. 95
Expanding and transforming, it will be found that the image of {U — F) is
Q'^r^U — — — v) and the second image of U which is ^ ^- ^ does not
revert to the form U.
As a simpler example we may take the covariant to a quartic, still of the
fourth order in the coefficients as before, but of the eighth degree in the varia-
bles. This will have for its root-dilferentiant
Xa" {ae — 4tbd -f 3c^) + (i{ac — b% ssiy^U+iiV.
Here TJ zna^ i~ -|- ^ — — (12(T4 — ScTitTa + (Tj), and for the purpose of
transformation is equivalent to
Hence, using /to denote "image of,"
/?7zr i- 1 J* + aV — ^ac¥ | = ~ F.
Again
V- (^ — ^
V 6 16/
= ^. { 3a^ + 3/3^ + V + 35' - 2 (a/3 + ay + a5 + i^y + /35 + y5) I
which, for purposes of transformation will be found equivalent to
~ I 36a^ + 132a^/3* — 48a*/3y — 144a^/3 + 24a/3y3 J .
Consequently
IV = -J- 1 ^c^e + llaV — Wb^c — 12a*firf + W |
= l^ 1 3o^ (a« _ 4i(i + 3c^; + 2 (i* — ac)* |
= 4(3 £^+2 n.
Let now ^ : ju be so chosen that
96
Sylvester, On Herrnites Law of Reciprocity.
This gives
fU
'w+(| + 4)''=-(^^'+''''''
or
64
t. e.
3^'
96
2?^-
X*
2
= 0,
96?.* = .
16
The two values of -^ derived from this equation are 6 and —. The
corresponding values of p will be 6 and — — • There are thus two definite
systems of ;i: /t/, and no more, w^hich will make XU -{- ^V self-conjugate and
it is obvious that there will be no other values oi 'a: (i which will make
for, P U and P V being determinate linear functions of U, F, we shall have a
quadratic equation for determining X : fij but the two values of 7.: fi which
make 2.U + /t/ F self-conjugate must satisfy this equation, and hence there can
be no others. Reverting to the preceding example of the type [6: 4, 4], we
have found
IU=^ U—— V
36 36
I(V-V)=^U
432
W
Hence
IV =
9
U
11
V,
and making
the equation for finding p will be
12
432'
__9^
432
432 ■ 432
whence
also, since
pi = —
1^
27
432
_ 11 _
432 '
b
P* = i44^
= 0,
_9^
432
(ij = p?.,
Sylvester, On Hennite's Law of Beciprocity, 97
we shall have
Ai y A>2 Q
(ii 28 ' H2
What intrinsic peculiar properties are possessed by the principal forms* is a
question as to which we are at present quite in the dark, as are we also with
regard to the general character of the equation in p. It were much to be
wished that some one would work out the case of a triadelphic type, as for
example the type of covariants of the 6th order in the coefficients and the
6th degree in the variables, to a sextic. It might be supposed from the two
preceding examples that the values of p are necessarily rational, but it will
be shown hereafter that such is not the case.
It is easy to see that the relation between any form belonging to a given
type of multiplicity 2 or 3 and its second image may be geometrically repre-
sented by means of a quadric curve or surface. Thus suppose the multi-
plicity is three, and that the three values of p are A. B. C. Construct an
ellipsoid or hyperboloid whose semiaxes are — _-: , — — , — - . Draw r any
\/ A w B wV
radius vector making angles a, ^ ,y with the principal axes, p a perpendicular
from the centre upon the tangent plane at the point where r meets the quad-
ric, making angles X, /t/, v with these axes. Then if
K (cos a U '\- cos /? F + cos y W)
be any given form of the system for which f/, F, Ware the principal forms,
K
— (cos W -\- cos /w F + cos V W)
pr
will be its second image. And we may say that, if a form lies in the direc-
tion of the axis of instantaneous rotation, its second image will lie in the
perpendicular upon the invariable plane: or more simply if by the direction
of a form ;i?7+|wF-f x^fFwe understand that of a straight line whose direc-
tion cosines are k%\ ^w and by its modulus ^7} + ^u^ + ^'^ we may say that
if a radius vector of the ellipsoid (or other quadric) represent the direction
and modulus of an in- or co-variant the corresponding radius vector of the
polar reciprocal to the quadric will represent the direction and modulus of its
second image.
* By a principal form (in general), as hereafter stated in the text, 1 mean one which is the reciprocal of its
first image in the sense that it bears a numerical ratio to its second image. The numerical qunntity by which
it must be multiplied to give the second image, I call a principal multiplier.
26
JW Sylvester, On Ilermites Law of Beciprocity.
The true nature of the reciprocity theorem, in the general case where i,j
have any values \vhatever, is now obvious. Let f/,, U^ i/^ be inde-
pendent forms belonging to the type [xv: i. j], whose multiplicity is §, and
l\, V.2 Fy as many forms belonging to the reciprocal type [iv: j, (]- We
may, by virtue of the transformation process, express each lU in terms of
linear functions of the forms V and vice verm^ so that each PU will be a
known linear function of all the J7's. For clearness sake suppose y = 3
and let
Now make
P (>M, + iiU, + rU, )=p{>.U, + uU, + vU,).
We shall have for finding p the equation
(a — p), b , c I
(i! , (/>' — p). ^ =(K
a" , //' , (^ — p)
and then the three systems of values of X: ft: v, which makes the second
image of >.Ui + ^U^i + rU^ coincide to a numerical factor pres, with itself,
will be rational functions of the respective roots. So, in general, when the
multiplicity of the type [««;; LJ] is y, there will be in general q special forms,
and no more, which have reciprocal forms belonging to the type [w; J, i], and
if the interchangeable elements, i,j are equal, then these q forms will all be
self-conjugate. It is conceivable that in certain cases the equation in p may
have equal roots; in that event each such equality would introduce a cor-
responding indeterminateness in the forms admitting of conjugates. For
example, if the multiplicity were 2 and the two roots of p equal, that would
signify that erery form belonging to the type would have a conjugate — a fact
analogous to an ellipse becoming a circle, or an ellipsoid a spheroid — and so
in general.
A form having a conjugate, L e, whose second image is a numerical mul-
tiplier of itself, may be called a principal form. If the multiplicity of the
type is y, there will be j.such. All but these will give rise to an endless
succession of images such that any ^ + 1 of an even -order (the form itself
included among these) will be connected by a linear equation. That the
succession is endless is clear from the consideration that if an image, say of
Sylvester, On Hermite's Law of Reciprociiy. 99
the (2p)th rank, is identical (to a numerical factor pres) with the form, we
have an e([uation of the qih, degree for finding the values of the systems of
multipliers X, /t/, v of U^ F, W\ therefore there are only q such systems, but
the systems which satisfy PF =. pF must also satisfy T^F:= p'JF, and conse-
quently there are none others.
To illustrate this, suppose
PU=aU+bV
PV=cU+dV;
then PU= {a' ^ be) U + {ab + bd) V
PV= {ca + ad) U+{cb-\-d^) V.
If now we put
a — p , b
c yd — p
to find the values of X : /u which make r^{XU -\- (iV) = p{XU + (iV) vfe have
{a — p) X + (?/u = .
In like manner, if we make
«^ + ftc — p , ab -^ bd
ca + ctd , eft + rf^
to find the values of A and M which make P(XU + [iV) = BiTiU + fiV),
we have
(a2 ^f,c—Ii)A'\- {ca + arf) M = 0,
and it will be found that
a — d
= 0,
= 0,
a — p= -^- _t ._ V (a — dy + Uc
a^J^hc — R= - ± ^^^ >v/(a — df + Uc ,
so that the values of X: fi and A : M are the same, and such we know a priori
must be the case.
It ought to be noticed that the method explained in the preceding pages
furnishes a complete solution of the problem following. Given any in- or co-
variant, say of the ;th order in the coefficients to a form Q of the tth degree,
to find the process of differentiation which performed upon the product
Q (^'ii i/i). Q (^2, y-z) Q (^y, i/j)
shall produce the j-partite-emanant of the in- or co- variant so given, and it
proves incidentally that every binary in- or co-variant may be represented as
a hyperdeterminant. To make this clear, let us call the above product,
100 Sylvester, On Hennite's Law of Reciin'ocity.
or rather that product divided by (Ilty, the j-ary norm of Q and denote
it by NQ. Again, let G be any given diflFerentiant to the type [w: j, i], say
G (p„ pg py) which is necessarily identical with
G {0\ (p2 — p,) ; (p3 — pi) ; (p3 — 1)} .
For p^ — p, write --— • — . — — and let the quantity so formed be
^ ^ dx, dy^ dy, da\
called the hyperdeterminant to G and be denoted by HG. Then if E be any
principal form to the type [w : i, j], of the multiplicity q and belonging to a
quantic Q, and G be its first image, we shall have
iHG){NjQ) = pF,
where p is one of the roots of a known equation of the qth degree in p. Con-
sequently, since any form belonging to the given type is a linear function of
its q principal forms, every such form may be expressed by means of the
hyperdeterminant
ll:y^{HG,)NQ,
Pa
the given form being supposed to be expressible by 2^^ * c^ t\^ where F\^ any
one of the q principal forms.
It follows from what has been shown above that in general from any one
particular given form belonging to a type of multiplicity q may be deduced
the {q — 1) others (by taking the successive second images) and thus the general
form obtained ; the exception is when the given form happens to be a linear
function of less than q of the principal forms. A further consequence is that
any in- or co-variant given in terms of the roots of its quantic may be con-
verted by explicit processes into a function of the coeflScients. Thus ex. gr.,
suppose that the multiplicity of the type is 3; call the given form ^o and the
successive second images B^^ i?2, i?3, R^, These latter will be all known by
the rule of transformation and we shall have -^4 a known linear function of
the three preceding forms, say equal to
aR, + /3i?2 + yi?3 ►
Hence if we put
-Bo = ^R\ + i^Ri + vR^ t
we must have
R, = ^R, + (iR,^v (ai?x + (iR2 + yRz) ;
hence
a a a
Sylvester, On Hermite's Law of Redprodty. •1()1..*
and thus jKo; given in terms of the roots, become known in terras of the coeffi-
cients of its quantic. And so in general, q being the multiplicity, (j + 1)
forms deduced from the given function of the roots will serve to determine its
value as a function of the coefficients. In fact by regarding ^o as ^ linear
function of the principal forms, it is easy to see it and all its successive secon-
daries (i. e second images) form a recurring series, the scale of relations being.
-Ro — 2 — ^1 + 2 -^ ^2 — 2 -^ ^3 + = ,
P P P
where i: p is the ratio of any principal form to its immediate secondary. Thus
Eq being given in terms of the roots and consequently E^ Eiy . . . . JS^y in
terms of the coefficients, Eq becomes known in terms of the coefficients and of
the quantities 2 — ,2 — ,....; these latter are identical with the quantities
P P*
previously mentioned and furnish the simplest means of forming the equatipn
in p, which (if we agree to call pi, p2, - . . . p^ the moduli of the several prin-
cipal forms Fly F2, . . . . F^^ L e. the ratios of their respective second images to
themselves) may be termed the modular equation for any given type.*
It might have been useful, had I thought of it in time, and may be useful
when the subject comes again under consideration, to treat a form and its
second image, in which the type is restored as antecedent and consequent^ and
to describe the first image as the aUernute form to the primitive, inasmuch as
we pass, by what biologists alternate generation, from one type to the other.
It has been shown, in what precedes, that the transformation by images at
each second step leads back to the original type, but, contrary to what might
have been supposed, does not in general imply the resuscitation of the indi-
vidual form.
The theorem of reciprocity has been seen to be, in its essence, a
theorem of differentiants, and ought therefore to admit of being proved by
means of the necessary and sufficient partial diflFerential equation to which
differentiants are subject. This may be done as follows. If we call e^^ e^
62, .... fr the successive elements to a binary quantic expressed in its wstom-
ary form, so that Sr is the coefficient of the term containing y^ divested of its
numerical binomial coefficient, and if we write
doL rf/3 dy
* But it will be better to adhere to the previous convention and to designate the p's as the principal multi-
pliers and the equation in p as the principal equation.
. ••
• • • • *
• • •• •- •
• » • •
• •. • • •
•• •••.
-.••..
. • •
•JOi' Sylvester, ()n IIerinitf\^ Law of Beciprocity,
where a, ,S y, . , . , are the roots of the quantic. it is very easily proved that
V^r — —r^r I.*
Let Cla''^y .... be any term in a given differentiant t\ the indices r, s, t . . .
being any whatever with no condition as to their being distinct from eacli
other, and let T(r, .h, f,.,..) signify the number of combinations comprised
in 1 ; also let C\ (r, /f, ^ ....). f^ f, f, .. . be called the image of the term
above written and G the image of F, /. e. the sum of the images of the several
terms in /'; where it must be observed that the e quantities do not necessarily
refer to roots the same in number or name as the roots a, .i, y, . . . . Now
suppose that we have any term, such as (jla'S'^y". ... in Ct] where U
refers to the ij^iven roots a, 3, y and means -,- + tt + ^i — [-•••• ^his
term must arise from terms of the several forms
A la' ' J" y" . . . . )
// la' 3"' 'y" ^ . . . y in K
C la' 3"' y ^ . . . )
corresponding to these there will be the images
AX (/ + 1, ///, n ) f' ^ f "* . t" I
IL\ (/. ?/« + 1, w ) t', . f "• '. t" . . . . > in G,
(\\ (7, m. w + 1, ) 8i . e" . e" ' . . . . ')
etc. etc
where G belongs to a quantic whose type is reciprocal to that of F, and it is
clear that the effect of operating upon F with U will be to give
Q = ApX (I + 1, m. n )(/+!) + BffX {L m + 1, // ) {m + 1)
•+ C^y (/. ^n. /i + 1. . . .) {n + 1) + etc
p being a number easily determinable, but which there is no occasion to
express. Again if Ihi. <•„,. f„. . . . be the correlative term in G, we have by
virtue, of the formula f/iE^= — rf^-i. where the operator fT' refers to the
roots of the quantic of reciprocal type,
(— )"i? zz AN (I + 1, m, n ) (/+!) + BN (/, m + 1, // ) {m + 1)
+ rX (L fn. n + 1, ) {n + 1) + etc
(Jonsequently, since on account of the identity J^ =: 0, we must have ^ =:
* In fact it may easily be proved by the ordinary rule for the change of one tyj«tcm of independent varinblcs
into another that, if a,, «,^, . . . « a, be the roots of (fq, f j, fji • • • • ^*\ ^t y)*»
V* d ^«-» d
Z T" = — ^ 9^9 -y T" •
da q — d€q
Sylvester, On Ilermit-es Law of Becip^rocity.
103
for every terra Q2a'. /iJ'". y" . . . . , we must also have if = p -^ ^ = and there-
fore, this being true for all the arguments e,. t^. f„. . . . , we must have UdzzO.
Hence, when any quantity jP, is a diflferentiant of a given quantic^ its image
(as defined in the text) is also a diflferentiant to a quantic of reciprocal type
to the given one. This is the simplest method of establishing the theorem,
but still the method originally employed in the note is valuable as serving
to establish the important proposition that every in- or co-variant of a binary
quantic is a hyperdeterminant.
I will proceed to show that for a system of two or more quantics of
degrees i, i\ ^", . . . . , we may pass from a covariant of the type [a? : i, j ; /', / ;
/", /;...] to one of the type [w : j^ i ; ?', / ; /", /;...] by taking its image in
respect to the quantic whose indices, /, ;, are to be interchanged precisely
according to the same rule as if there were no other quantic present. As
regards the law of reciprocity, a combination of quantics is analogous to a
mixture of gases, according to Dalton's view, each playing the part, as it were,
of a vacuum in respect to the other.
Let [m' : /, j ; i\j' ;....] be the type, [w : j, / ; i j\ ;....] one of the anti-
types, (go, ^1, f-i, . . , Bj\ X, yy the general form of the jth degree, a, /if, y, . . . •
its roots when fo =^ 1- Let >7^ = ( — )'^f^;.then, since
2— f, = — rf, _,
da
da
yjr = yyir -
Let D be any diflferentiant of the given type, a, b, c, . , . the roots of the
quantic of degree /, a', b\ c', . . . the roots of the quantic of degree /', with
the usual convention as to the leading coeflRcients becoming unities. Let
2«'6"*. . . . , ^a'^'b""'. ... 2 ... . be the arguments of any term in
da da'
say UI), then the coeflficient of the term last written will arise from operating
with U upon
(^.7+^.^.+
)A
A- la! + '.ft" . .
. . 2a"'
ir'
. eLc. . .
+ B- In! . ir ' '. .
. . 2«"'
ir'
. etc. . .
+
1
+ A' la! . //" . .
. . la!'' '
I Jj.m-
. etc . .
+ B' la' .b" . . .
. . la!''
A""'
. t^l/C>. . .
etc. . . .
etc. . .
>
J
w
104
Sylvester, On Uermites Law of Reciprocity.
and the value of the coeflScient will be
^ (/ + 1) .V (/ + 1, m, ) .V [t m' ) . . .
+ Ji {m + 1 ) y (/, m + 1 ) .V {f. m\ ) . . .
+
+ A' (Z' 4- 1) -V (/, m, ) X {f+h m\ ..)...
4- B' (m'+ 1) N (L III ) N (^ m' + 1, . .) . . .
+
~ N (/, m^ . . . .) N {t, m\ . . .) ...
To these feeders or contributory terms will correspond, in the image,
AN (I + 1, in ) Vi ri' 'Km • • • . 2«"' . Z/*^' . . .
+ BN (/, //i + 1 ) 17/ .>:«.,.... Srt^ .ft""' ...
+
]£«"' ^ ^ 6'-'
2«"' . ft'-'
+ A'jX (/, m, ) >:, .y;^ ...
+ /^'JV^(/, m ) yjf . >:« ...
+
and it is obvious that by operating upon this with the ?7 corresponding to its
roots we shall obtain the argument >;,. 17^. . . . Sa"*. ft'-', etc. . . . aflTected with
the very same coeflScient as that above written, except that in its denominator
the factor, JV^(/, m, . . . .), will not appear. Hence, when /> is a differentiant
of the given type, its image (obtained by expressing the / set of coeflScients in
terms of roots and then replacing every power, p', of any such root, p, by >7^,
leaving all the other coeflScients unchanged) will also be a diflferentiant of the
type transformed by interchanging i with its conjugate j.*
When there is but one quantic the eflfect of substituting Sg instead of yj^
will evidently only be to introduce a common factor ( — )* into each term,
which is immaterial and we may accordingly in that case reflect p* into €g.
Of course, in the general case, if all the letters i are simultaneously inter-
changed with the letters j, a similar conclusiou follows.
* Thus tbe rule of images for passing from a differentiant of a given type belonging to a single quantic to
x>ne of the opposite type is extended to tbe case of passing from a differentiant of a given type belonging to a
/tystem of quantics to any associated type, t. e. to any lype in which one or more of tbe numbers i chosen at dis-
cretion is or are interchanged with the corresponding numbers j, and it will presently be seen that this Implies
the extension of the rule without any alteration from differentiants or invariants to covariants of a quantic or
system of quantics. In Note A. it will further be shown that for any inversions whatever (or, to speak more
accurately, for any cycle of inversions leading back to the original type), although the principal multipliers
change their values as the cycle of inversion changes, the principal forms themselves remain the same^ — a sur-
prising conclusion but very easily proved. In other words, however many quantics there may be in the parent
Mystem, there is never more than one single setx)f priiicipal forms of derivatives to it of a given type. A cycle
of arbitrarily intercalated pairs of reversals (here of successive t's and j's), by which a type returns to itself,
comes under the category, *' Verschlingung," or ** Knotting '' of Oauss, Listing and Tail.
Sylvester, On Hermite's Law of Reciprocity, 105
As an example, let us take the two quadratics,
aj^ + ^bxy + ^y?
ax^ + 2(ixy + y/,
their resultant, {ary — caf + 4 (a/3 — ba) {c(3 — 6a), belongs to the type
[4 : 2, 2 ; 2, 2] which is its own reciprocal whichever of the interchangeable
elements, we permute. This resultant, treating a as unity, will be equal to
(op? + 2/3p, 4- y) {apl + 2/?p, + y)
= aV?p^ + 2^a (p?p2 + pip^) + 4/?*^pip2 + ay (p? + p?) + 2^y (p, + p^) + f
the image of which will be
ahl — 4a/3f if2 + 4/?^e? + 2dyfof2 — 4/3yfofi + yM ,
or as we may write it,
aV — 4a^bc + 4/326^ + 2aya^ — 4^yab + ay ,
which is {ca — 26/3 + o/yY , the square of the well known connective. Again,
if we combine aar^ + 3bary + Scsy^ + d/ with aor + /^y? we have the invariant
a/3^ _ 3fta/32 + 3ca2/3 — da^ , say /,
belonging to the type [3 : 3, 1 ; 1, 3]. Write a zz 1 , ^ = — p ; this becomes
— ap^ — 36p^ — 3cp — d ^
of which an image, say /, belonging to the type [3 : 3, 1 ; 3, 1],
ass — 36^2 + 3cei -«— deo
is the connective of f aor^ + ^bx^y + 3cxy^ + dy^ )
[6^ + SsiX'y + 362^/ + e;^3. J
Similarly
(a^d — 3a6c + 26')/3^ + + {d:'a — 3dbc + 20^)0?,
say /, belonging to the type [6 : 3, 3 ; 1,3], will have for a reciprocal
{(^d — 3abc + 2b') 63+ {dT^a — Sdbc + 2c') so,
say J, belonging to the type [6 : 3, 3 ; 3, 1] , The graph of / will be that of
Fig. 41 and the graph of /, that of Fig. 42, where I use £ and G (the initials
of boron and gold, instead of Au for the latter) and H (the initial of hydro-
gen) to represent the algebraical atoms (i. e. quantics) of valences {i. e. degrees)
3, 3, and 1 respectively. Prefixing 2 to the / graph and substituting G^^ Go,
Gsj the three roots of Gj for H^ H\ H" and B^, £2^ £z for B^ B, Br we obtain
2 {B,-B,) {B,-B,) {B,-B,) {B,-G,) {B,-G,) {B,-G,),
which by inspection is the root representative of /, and prefixing 2 to the /
graph and substituting //for G, we obtain in like manner
2 {B, - B,y {B, - B,) {H- B,) {H- B,)\
as the root representative of /.
Vol. L— No. 2.-27
106 Sylvester, On Hermite^s Law of Reciprocity.
It may be observed that Fig. 43 is, algebraically speaking, a pseudo-
graph of /, for its reading would give zero for the value of /.
It follows as an immediate consequence from the preceding extension of
the law of images to quantic-systems, that the rule for deducing the first term
of the reciprocal to a covariant from that of the covariant itself by writing r^^
for a'' holds good as a rule for deducing each term of the one from the corres-
ponding term of the other. To see this we have only to recall that every
covariant to a quantic or quantic system may be regarded as an invariant of
a new system containing the given quantic or system augmented by a linear
quantic whose coefficients are y and — x.
Note A to Appendix 2.
Completion of the Theory of Principal Forms.
In the case of a derivative from a system of k parent quantics, it at first
sight would seem that since reversion (the act of forming the second image,
or process, as we may term it, of double reflexion) may be eflfected in regard
to each system of coefficients separately, the method in the text ought to
furnish in general k distinct systems of principal forms, but this is a mere
mirage of the understanding which disappears as soon as the question is
submitted to close examination. There is always an unique set of ft forms
ifi being the multiplicity of the type) which revert unchanged (barring a
numerical multiplier) whichever system of coefficients undergoes double
reflexion. But a caution is necessary for the right interpretation of this
statement. U, V, W . . . . may be the principal forms in regard to one
set of coefficients, ?^U + (iVj W . . . ^ or 'kU + (iV -\- vW . .., where X, /z, v
are indeterminate, in regard to some other. In any such case we may
still say that U^ V^ W . . . . \^ the principal system in regard to both sets
and so in general. We have an example of this if we take any covariant
to a single quantic Q and translate it into an invariant of Q and a linear form
L, It Uj V, W . . . are principal forms in respect to Q, XU+ (iV+vW'{' . . .^
(/. e. the absolutely general form of the type) may be easily shown to undergo
reversion in respect to L tinaltered. Uj V^ W . . . . may consequently still be
seen to be a principal form system in respect to Q and X, as each of these
quantities is unaltered by reversion in respect either to Q or to L.
/
Sylvester, On Hermite^s Law of Reciprocity. 107
Suppose now a diadelphic system of which ?7, V are the principal forms
qua one set of coefficients. Let R denote a reversion qua this set, i?' qua
some other set. Let RU=aU, RV= bV and suppose RU = aU+^V.
Then- RRU= aaU + b(3V and RRU= aaU+ b(3V.
But by the nature of the process of reversion RR = RR ; hence a^ = b^.
If a zz 8, every linear combination of £/, F is a principal form qua R, Hence
the principal form qua the R set, is such for both sets. But if a is not equal to 6,
we must have /3 = 0. Hence U will be a principal form qua R as well as J?,
and the same will be true of V. For if "*
R^V=yU+SV
RRV=ayU+b8V
RRV= RbV= byU+ bhV.
Therefore ay = by and y =z 0. Thus U, V will each of them be common as
principal forms to each set. I have gone through the same somewhat tedious
process of proof for triadelphic forms and with the same result. The very
beautiful conclusion follow? that whatever the multiplicity of a type may be
and whatever number of sets of coefficients it involves, there is always a single
system of principal forms common to all the sets.*
Note B to Appendix 2.
Additional Illustrations of the Law of Reciprocity.
Acetic aldehyde contains two atoms of carbon, one of oxygen and four
of hydrogen .f It thus corresponds to the quartic co variant of a quadratic and
* Suppose there are k quantics in the parent system and that a derivative type fi is given. Each simple
inversion of a pair of permutahle indices (i^j) will give rise to a distinct principal equation ; there will there-
fore he k such equations. Let p he a root of one of these, a a root of any otHcr. Then a principal form may he
expressed as a linear function of any [i independent special forms connected hy goefficicjits.Y^hich are rational
integer functions of p. Hence a may he found as a rational function of p; but in like manner p may he found as
a rational function of a. Hence p, a must be related hy an equation of the form
Apa + £p -|- Or + Z> = 0,
and thus we see that all the k principal equations are homographically related, i. t. that each may he obtained
from any other by a substitution of the form
_ Co+D :
^~ Aa+B'
In a word, the multiplicity fi (whatever the diversity k) determines the number of principal forms; and the k
sets of principal multipliers are given by k algebraical equations of the /ith degree, homographically transform-
able into one another.
f I originally took chloral as the subject of this investigation, being interested in examining its algebraical
constitution in consequence of having had personal experience of its use as an escharotic. But for greater sim-
plicity I have substituted acetic-aldehyde of which chloral is a third emanant, three hydrogen atoms of the
former being replaced by three of chlorine in the latter.
108 Sylvester, On Hermite's Law of Reciprocity.
quartic, linear and quadratic in respect to the coeflScients of the first and second
respectively; such a form exists algebraically (Higher Algebra 3d ed., p. 200)
and may easily be proved to be monadelphic. Let us treat it as an invariant:
if we were to take for its graph a triangle of which C^ C, were the apices
and attach two atoms of hydrogen to each C, the permutation-sum of the
product of the differences of the connected letters is zero; this then is a
pseudograph. A true graph of it is given by the figure
H'C'O'H
■
where each single dot between two letters means a single bond and the two
dots between the upper and lower C^ stand for a pair of bonds between
them. This belongs to the invariantive type [4, 2; 2, 1; 1, 4: 0], the com-
plete reciprocal to w^hich is [2, 4; 1, 2; 4, 1 : 0]. The constitution of the •
latter in terms of the roots is found from the above graph by writing for C,
C for 11 and // for and is accordingly
2 {0—Oy {0— C) {O—O) {O— C) {O — H) {H— C),
where the factor {0 — Of may be put outside the sign of summation. We
may therefore take for ite graph a detached molecule of oxygen + a molecule
of formic acid, which latter contains two of oxygen, one of carbon and two of
hydrogen.
EC OH
» :
will be a graph of it, from which, turning into C^ II into and C into E
we obtain
^{C— Of {C — E){0'— O) {O' — E) {E— 0)
as the value, in terms of its roots, of the algebraical equivalent to acetic
aldehyde. The graph for formic acid, it may be noticed, exists algebraically
(Higher Algebra, p. 300).
Instead of the dissociated molecules of oxygen and formic acid, we may
exhibit them combined in the graph
C'OO'O'E
m
which will give another form to the value of the reciprocal in question, viz.
2 {C— H)\H— 0){H— C'){C — C^iC — Cr'){0' — 0)
Sylvester, On Hermite's Law of Reciprocity, • 109
which, not being zero and the type being monadelphic,* must be in a pure
numerical ratio to the sum above written.
Chemistry has the same quickening and suggestive influence upon the
algebraist as a visit to the Royal Academy, or the old masters may be sup-
posed to have on a Browning or a Tennyson. Indeed it seems to me that an
exact homology exists between painting and poetry on the one hand and mod-
ern chemistry and modern algebra on the other. In poetry and algebra we
have the pure idea elaborated and expressed through the vehicle of language,
in painting and chemistry the idea enveloped in matter, depending in part on
manual processes and the resources of art for its due manifestation.
A peculiar case might possibly arise in applying the theory of principal
forms to a self-reciprocal type \w : i, i\ which it is proper to mention. For
greater simplicity suppose the type to be diadelphic and let Jf, N be forms of
^ the type which satisfy the equations
the Jlf and ^^have tacitly been defined to be the principal forms for such a
type. Now in general this definition merges into and is coincident with the
definition of principal forms for the general case, viz., that PJ/and PJV^must
be multiples of M and N and the latter condition might be substituted for
the former. But this is not always true, for if p + p' = 0, wc shall have
and consequently,
P {M + :>,N) = p2 (Jlf + XiV^) ,
so that if we were to follow the general definition the principal forms might
become indeterminate, whereas by following the definition special to the self-
reciprocal case they are determinate. Thus ex. gr., suppose that P, Q, two
particular forms of the type, satisfy the equations
/P=pQ, IQ = aP;
* As an exercise the reader may satisfy himself that this type is monadelphic by the direct application of
the rule for finding the multiplicity. It corresponds to a quadratic covariant of the type [2, 4 ; 4, 1 : 2], which
18 the same (introducing the weight — — ' in lieu of the degree) as the type [6: 2, 4 ; 4, 1] and has
the same multiplicity fi by the law of reciprocity as the type [6 : 4, 2 ; 4, 1], viz. the difiference between the
number of modes of composing 5 and of composiog 4 with two of the numbers 0, 1, 2, 8, 4 and with one of a
dUtinei set of the same numbers. The arrangements for the weight 5 will be 4. 1 : 0, 4. 0: 1, 8. 2 : 0, 8. 1 :
1, 8. : 2, 2. 2 : 1, 2. 1 : 2, 2. : 8, 1. 1:8, 1. : 4, and for the weight 4, 4. : 0, 3. 1 : 0, 8. 0: 1, 2. 2 : 0, 2. 1 :
1, 2. 0: 2, 1. 1 : 2, 1. 0: 8, 0. : 4. The numbers of the combinations in the two sets of arrangements are respec-
tively 10 and 9. Hence // = 10 — 9 == 1, or the type is monadelphic. The same result of course follows from
the known fundamental scale for a quadrobiquadratic system,
28
110 Sylvester, On Hermite's Law of Reciprocity.
the principal forms will then be
Vff'P + VjT (^ and Va P—\^fQ,
and the two principal multipliers become Vpa and — Vpa, so that the prin-
cipal forms according to the general definition would be indeterminate, but
according to the definition proper to self-reciprocal forms strictly determinate.
Let us, as a final example of self- reciprocal type, consider the type [10:
o, 5] which is the same as [5, 5 : 5] and corresponds to the covariant of the
5th order in the coefficients and of the 5th degree in the variables to a quintic.
This is diadelphic, as may be found by consulting the table of irreducible
forms for the quintic, which will show that it can arise only from the multipli-
cation of the parent quantic itself by its quartinvariant or from that of the
quadratic quadri-covariant by the cubic cubo-covariant or from a linear com-
bination of the two products. But without this, the same conclusion may be
arrived at by direct calculation of the value of (10 : 5, 5) — (9 : 5, 5) and the
multiplicity will be found to be 18 — 16, or 2 as premised. Let us take as
our special forms, ^
p - (^ae — Ud + Sc") {ace + 2bcd — a(P — c^ — b'e).
q-a {d\f — lOabef + Aacdf + 16w^ — Uad'e + imdf + 9AV —'12b(ff
— 76 bcde + 48 i(f + 48(rV — 32c'd'),
where — is the quartinvariant J given by Salmon, p. 207, (3d ed.), being
a
in fact the discriminant of the quadricovariant whose root-differentiant is
rie — 4bd + 3c^. Call a, /?, y, 5, e the five roots of the quintic and make
a = 1, Q contains the term /^ which is the image of a^(i^ which can only
arise from combinations of the coefficients into which rf, ^, / none of them
enter. But all the terms of Q contain d, e^ or/, moreover P has no term con-
taining /'^, therefore IQ, does not contain Q, but is simply a multiple of P.
Again ce^, which enters into P, is the image of combinations of the form
a^iSy^j and the only term in Q which can give rise to such combinations is
— 32(?2rP, or
and each such combination will have unity for its coefficient and their number
is 30. Hence
70--^'^^ P-- ^1 P
^ ~ iOOOO 125
Sylvester, On Ilermite's Law of Reciprocity. Ill
Again, Q contains — lOi^/, and bef'i^ the image of such root-combinations as
aJ^^^Y (60 in number) the only terms in P capable of producing which are
10b(?d and — 3c« or -^ 2a (2a/5)^ 2a/3y — :rj^^r^ (Sa/?)^ And 6^/ does
not appear in P, hence one part of IP will be
/ 60 3- 5-60 \ . 3^
V — 50000 "^1000000/ ^' *^'' lOOOb^'
Again, ce^ is the image of such combinations as a^^y^ (30 in number) and the
only terms in P giving rise to such are — 3tr^ — ^^ccP + lObcM — 3c*(f ; — 3c*
is — jQgooo ^^""^^^^ ^^^ ^^^^ ^^^'^ ^^^^ ^^ ■" iooooo ^^ ^^ ^^'' ~ ^*'^^ ^®
"26500^^''^'^^''^^ (2a/3y)^and will give rise to -||)^^^in /P; lO&^rM
is — -— r2a(2a/3)^2a/5y and will give rise to -pr.^^,- e^ in IP\ — 3(r^(f is
— fmm (^*^)^ (S«/^y)^ a^d ^^^^^ gi^® ^i^^ ^^ ~ TOOOO ^^ ^^ ^^* Hence the
total coefficient of ce^ in /P is
9 _ 12 21^ 9 __90 — 96 + 210 — 45 __ 21
500 625 "^ 500 1000 6000 ~ 5000 '
and consequently, since P contains the term ce^ and Q the term lQc(?, if
a 3 16 21 ,, , . 3
% TT^, =■ — i'A?^ > so that 6 =
10000 ~ 5000 ' ~ 5000 '
and therefore
jp — _3_ p ?__ n
~ 5000 10000 '
and thus the equation for finding the principal multipliers p is
3 3
5000 *^' 10000
12
-125' -^
= 0,
"^'^^ P = iOOOO' i-320, -a!=^-
112
Sylvester, On Hennite's Law of Reciprocity.
Thus (T^ — 2cT — 320 = 0, the roots of which are irrational. I have thought
it advisable to set out the work in this example with some explicitness in
order to remove an impression that might otherwise arise from the examples
which precede, that the principal multipliers and consequently the principal
forms, for self-reciprocal types, necessarily contain only rational numbers.
The work is very much longer for the case of non -self- reciprocal types.
The simplest example of such that presents itself to my mind is that of the
sextinvariant of a quartic and the quartinvariant of a sextic, for either of
which the type is diadelphic. The discussion of this case forms the subject
of the annexed Note, for all the calculations of which I am indebted to the
labor and skill of Mr. F. Franklin, Fellow of Johns Hopkins University.
For the sake of brevity the steps of the work have been suppressed and only
the final results set out.
Note C to Appendix 2.
On tJie Principal Forms of the General Sextinvariant to a Quartic and Quart-
invariant to a Sextic.
Let
L =. {ae
M =
8
a,
b,
c
b,
c,
d
G,
d,
e
P={ag
m + 30' = [~3 S (a - ^Y (y - hY'\,
* = {ace + 2bcd — ad"" — b'e — cj
1
(j^ =
ebf + 15ce — lOdy = [
a, b, c, d
«)]',
b, c, d, e
c, d, e, f
d, e, f, g
1
2*-3-5
aceg — acf^ — ad'^g + 2adef
— a^ — Veg + **/* + ^cdg
— 2bcef— 2bdY+ 2bde' — &g
2(a-/8r(y-«r(^-<?>r] ,
2»-3''-5
2 {a-my-^)\^-^y-^^\^ {a-^)\y-V)\,-^Yl
*M. Faft de Bruno, in the tables at the end of his **Thtorie des Formes Binaires,'' designates Q and
Z (a — /3)^ (7 — ^)^ (^ — ^Y ^y ^^ same symbol /^ ; a misleading circumstance which gave Vise in this instance,
Mid might in others to a large amount of useless labor. As can easily be seen from the above, the true value of
2: (a — /?)* (y— <J}* (c — 0)* is 120 (7IP+ 900 Q)
= 120 (71aa^» — 862a6/y + 8080a«^ — 9006»<^ — 2820arfV + 18006crf^ — 900c»^ — 900ac/« + 84666y«
+ ISOOorfe/ — 146806c«/ + 6720^*/ + ISOOc^rf/ — 900o«» + 18006rfe« + 16876c««« -• 24000cd»« + SOOOrf*). It
should also be observed that in the expression for Q (the catalecticant) given in the same table, the signs of the
terms — 2M'/ -|- 26<(e> have been interchanged.
Stlvesteb, On Ilermite's Law of Reciprocity.
I
or
IQ =
6»
9L — 142M
PI - 7614/v + 238683f „,.-_ 201Z + 21623/
Id order that XL + [iM shall be a principal form we must have
(7614 — 2"yo'p) X + 20V = .
23868X+ (2162 — 2"-3*-5V)fi = 0,
I 7614 — 2"-3^-5»p , 201 I _
I 23868 , 2162 — 2"-3*-5^p l~ '
or, putting a = 2**3*-5'p ,
ff* — 1222(T + 182250 = ,
where it may perhaps be worth noticing that the last term is 2'3*-5' and the
coefficient of the second term 2 13*47. We obtain from this equation
_ 611 j^ V 191071 ^
** ~ 2«-3*-5*
The principal forms in L and M will then be found to be
2017, + {— 2726 + 8 V T91071) M , 201L + (— 2726 — 8 V 191071) M ;
and those in P and Q
IQIF + (— 11436 + 24 V 19i07"l) Q, lOlP + (— 11436 — 24 V 191071) Q.
Or, if we please, the principal forms in the two cases may be taken as the
factors of 201L^ — 54o2LM — 23868il/^ and 101i« — 22872^^ + 205200Q^
respectively.f The question, what reduced quadratic forms can appear in the
theory of diadelphic types, may one day or another become the subject of
dpriori investigation and form a new connecting link between the Calculus
of Invariants and the Theory of Numbers. The linear functions of L and M
and of P and Q^ corresponding to the reduced forms of the above expres-
•Tlie number under the radical Hign ii, 1 believe, a prime number, but I have not within rcaoh the tables
neceuary for verifying this. Professor Newcomb, by an eiceedinglj ingcnifiua combination of a table of
■quarea with Crello'a table of multipliers, (a real stroke of genius,) was able to ascertain by an inspection (the
work of a few minutee) that 1 9107 1 , if not a prime number, matt contain a factor not greater than a certain
moderate aised integer (187 if mj memory serves me right) which reduces the trials necessary to be made to a
Tery small compass.
f These are reducible to
(201,68, — 60800 J i',j»fj», (101, — 23, — 1089667 (P', Q)', where i' = Z. — 14if, /"= P — llSV
Sylvesteh, On Hennite's Law of Seciprocifyf.
aions might perhaps be ternied the principal rational forms of the two types
respectively.
It may be well to notice that if rU=pir, then P-IU:= IPU= pIU,
aiid consequently the principal forms for two reciprocal types are images
respectively of one another, and the principal multipliers are the same for
the two systems.
Note D to Appendix 2.
On the Probable Relation of the Skew Invariants of Binary Quintics and Sextics to
one another and to the Skew Invariant of the same Weight of the Binary Nonic.
The law of reciprocity extended, as it has already been in these pages,
to systems of quantics, admits of an additional important generalization.
We know that Regnault's law of substitution holds good for algebraical
forms, and in fact when transferred to the algebraical sphere becomes iden-
tical with the method which I believe I was the first to employ (now familiar
to algebraists through the use made of it by Professors Clebsch and Gordan)
to which I gave the name of emanation, (Faa de Bruno, p. 198).
The principle, stated in chemico-algebraical language, is that in alge-
braical compound any number of atoms of a given valence may be replaced
by the same number of new equi-valent atoms. [In algebra it is essential to
lay a peculiar stress on the word new, for if the substituted atoms should be
homonymous with the remaining atoms, there is a possibility of the trans-
formed compound reducing to zero. As for instance in the algebraical com-
pound off — «tb (the representative, say, of potassic iodide), if the atom of
potassium should be changed into another of iodiiie, (or vice versa), the com-
pound, viewed algebraically, would disappear].
The law of reciprocity as I have previously given it, translated into
chemieo- algebraical language amounts to saying that the total number of
atoms of one kind (say m n-valent of one kind) may be replaced by «
m-valent atoms of another kind ; but by applying the rule of substitution
first and then that of reciprocity we may see that the condition of totality may
be done away with and the proposition reduced to the simplified form that in
any algebraical compound m n-valent atoms may he replaced by n m-valent ones.
Whether this law has any application in the chemical sphere, I must leave to
chemists to determine.
Sylvester, On Uermite's Law of Reciprocity. 115
In addition to the well known fact that a quintic possesses an invariant
of the 18th order and a sextic, one of the 15th order, having obtained a com-
plete scheme of the irreducible invariants for the binary quantic of the 10th
degree, I was put in possession of the new fact that this last form possesses
an invariant of the 9th order and consequently that the nonic possesses an
invariant of the 10th order.*
Now the weight of each of these skew invariants is the same number 45,
and I was thus led to suspect that they coexisted in virtue of some secret
connexion. What that connexion is 1 think that I am how (very unexpect-
edly) in a position to explain and to show (with a high degree of probability)
how the values of these three invariants may be actually deduced and calcu-
lated from one another. This follows as a consequence of the combined laws
of reciprocity and substitution otherwise called emanation. For suppose we
have an invariant of a quantic of the mth degree, of the order np in the
coefficients. By the principle of emanation we may transform this into an
invariant to a system of n quantics, each of the degree m and of the order p in
each set of coefficients, and by the generalized law of reciprocity this may be
again transformed into an invariant to a system of n quantics, each of degree
p and of the order m in each set of coefficients. If now finally these n quan-
*I have calculated, with the kind assistance of Mr. Halsted, the expression in its canonical form of the
generating fraction to a binary quantic of the 10th degree. The coefficient of m in this fraction developed,
represents the number of parameters in the general invar'ant of the mth order of the given decadic. Its
denominator is
and its numerator is the rational integer function
l + 2^«+ +2<** + <*»,
the luccessive coefficients being
1, 0, 0, 0, 0, 0, 2, 0, 4, 2, 7, 6, 16, 18, 16, 26, 22, 81, 34, 40, 41, 47, 46, 49, 48, {49, 46, 47, 41, 40, 84, 81, 22, 26,
16, 13, 15, 6, 7, 2, 4, 0, 2, 0, 0, 0. 0, 0, 1,
showing that the primary fundamental invariants are of the orders 2, 4, 6, 6, 8. 9, 10, 14. and that (by the law
of ** Tamisage " anglice siftage) the secondary (or as they might be better termed the auxiliary) ones are of the
orders 6, 8, 9, 10, 11, 12, 18, 14, 16, 17 taken 2, 4, 2, 7. 6, 12, 13, 18, 21, 11 times respectively. Any other
invariant of the decadic can be represented as a linear function of a limited number of combinations of the
secondaries, having for its coefficients some combination of powers of the primaries.
Suppose that the same numerical order occurs among the primaries and secondaries, as ex. gr. 6, which
occurs twice among the former and twice among the latter. This will indicate in the first place that, calling A
and B the quadric and quartic invariants, the general sextic one will be of the form
X^» + fiAB + VjQi + v,Q, + V3Q, + v,q,
and that any two independent special values of ViQj + v^Qj + ^'sQa + ^'4^4 "^^7 ^® taken as prinaaries and
any other independent two as secondaries, and so in general ; I mention this to prevent the false suggestion,
which might otherwise arise, that the secondaries and primaries are different in internal constitution. This
remark receives a beautiful illustration in an algebraical theory (recently developed by me) of chemical isom-
erism, which gives rise to a generating function precisely similar in character to that applicable to in- and co-
variants and is subject to a similar law of interpretation, graphs taking the place of algebraical forms, and
atomicalet and the numbers of grouped atoms, of degrees and orders.
•
$
116 Sylvester, On Hermite's Law of Reciprocity.
tics, be all made identical with one another, then the transformed invariant,
provided it does not vanish^ becomes an invariant of the order m?i to a single
quantic of the degree ^, and accordingly we may pass in certain cases from
the type [m, np: 0] to the type [j?, mn: 0]. So in all probability we may
pass from the type [5, 18: 0] to the type [6, 15: 0] and to the type [9, 10: 0].
As there is only one invariant of the type [6, 15 : 0], or of the type [9, 10: 0],
it follows that, if the passage from type to type is real and not nugatory, the
three invariants of these second types may be deduced, any one from any
other, by the explicit processes above described. There is nothing at all
doubtful in the course of the transformation except what arises from the
possibility that in the last step of it the eflfect of rendering identical the
different sets of coefficients — i. e. of finding the counter-emanant, so to say, of
the invariant containing n sets of variables — may be to render the whole
expression null. This of course would happen if we attempted to pass from
the type [5, 18 : 0] to the type [3, 30 : 0], or to the type [2, 45 : 0], which we
know are void of forms. But there is no reason why we should expect this to
happen when we pass from the given type to other types known to contain
one or more forms. It would require no impracticable amount of labor to
actually verify the fact of the transformation being effectual between the skew
invariants of the sextic and quintic forms. The survival of a single known
term in either of them, in the process of attempting to deduce it from the
other, would be sufficient to establish the effectualness of the method, and
that it will be found to be effectual, for reasons too long to dwell upon here, I
scarcely entertain a doubt. The process to be employed may be summarily
comprehended under the three rubrics of diversification, reciprocation and
unification. The first is one of differentiation alone ; the second involves the
expansion of functions of the coefficients of an equation in terms of roots and
the substitution of yji for a* ; the third consists merely in replacing distinct
sets of letters (>;) by a single set. In practice the two latter processes would
be of course combined into one. It will be instructive to consider some
simple example of this method of transformation of types.
Let us take {ac — b'^y regarded as belonging to the type [2, 6:0]. I
shall show how to pass from this to a form of the type [3, 4:0]. Taking a
third emanant of the given form, i. e. the result of the operation upon it of
1-2^ (^'^a + *'^«)', we obtain
{a(f + (tc — 2hl/y + 2{ac — V) {aH/ — 6'^) {ad + afc — 2hh').
Sylvester, On HermMs Law of Redprocity, 117
*
Let us call a, /3, a', /?' the roots of the two forms [1, i, c], [1, 6', (f\ respectively ;
then the emanant last found (multiplied by 8) becomes
(2d/3 + 2d'|3'— aa' — a/3' — /3a' — /3/30 (2a/3 + 2a'/iJ' — oa' — a/3' — i3a' — /^/3' '
+ ^r=:;5^-^7^'').
After performing all the multiplications and introducing the zero powers of
a, a', /3, ^' in such terms as do not contain one or more of these letters, all
that remains is to substitute
a®ii:a'^ = ^«zz^'«= a,
a =z a' = /? =z /3' = — 6,
a? = a'2 = (3^ = ^'^ =z c,
a« zz a'« zz /33 zz |3'^ zz — rf,
the letters a, ft, (?, (Z for greater simplicity being used instead of foj ^u ^2? ^s? *'• ^.
>7o> — >7ij >72? — >73- The result will not vanish. To show this consider the
group of terms which change into aH'^. These are the binary combinations of
a^ a", /3^ /3'\ 2a/3 and 2a'/3' in the first factor give rise to 8a'/3^ 8a"|3'' and
the remaining four terms to — 2aV, — 2a^/3'^ — 2/3'V^, — 2/3^/3'^ respectively.
Hence the term a^rf^ will survive with the multiplier 8 + 8 — 2 — 2 — 2 — 2,
i. ^., 8. So again the only terms introducing cu? will be the ternary combi-
nations of a^ a'^ /3^ /3'^ 2a/3 and 2a'^'-will be found to produce as many
positive as negative terms of this kind, but — aa' will produce 4aV^/3^
+ 4a^/3'^/3'^ giving rise to 8ac^, and as the same will be true for — a/3', — /3a',
— /3/3', we see that 32e3^ will emerge in the result. Hence the given inva-
riant becomes converted into
{aH"" + 4«^ + ) ,
i. e. the discriminant of the cubic whose type is [3, 4: 0] as was to be shown.
I think it is little doubtful that wherever there exist forms contained
under each of two types, the product of whose rank and order is identical,
we may pass from the one to the other by means of the combined processes of
emanation and reciprocation, as in the foregoing example.* [The case is
» Call (62 -_ ac)8 = A, a^d^ + 4ac3 4. =5, a^Sa + h^ih + c^^c = E, ada' + bdi/ + cSff + d6^
=:zH-^, Then it follows from the text that
B = ^H-*IE^A ,
.where it may be observed that E^A is diadelphio, for it will be proved that (6 : 8, 2 ; 8, 2) = 16, and (5 : 8, 2 ; 8, 2)
a= 14, so that any form whatever coming under the same type as E^A is a linear function of (ac' + aU — 266') •
and (cLC^ — a^c — 2hb^)(ac — 6*)(a'c' — h'^)^ say L and Af, (whose difference, L — Af, is \E^A)^ and operated
on by H~ */ would produce a multiple of B (whose type is monadelphic) with the sole exception of XL — 2AAf,
the result of operating upon which would be zero. Similarly we may see that in any given case the chances are
infinitely in favor of the expectation that the process will not be nugatory by which it has been shown we may
pass from one known type [m, np : 0] to another known one [f>, nm : 0].
80
118 Sylvester, On ClehscKs Theory of the ^^ Einfachstes System^^^ &c.
much the same as with transvection. That process may produce a null form,
but any actually existent form may be produced by it and exhibited as a
transvect] To pass from Hermite's to Cayley's skew form, we must first by
emanation change [5, 18 : 0] into [5, 6 ; 5, 6 ; 5, 6 : 0] and then this latter
into [6, 15, 0] ; by means of the process last exemplified. • -
APPENDIX 3.
On Clebsch's Theory of the *' Einfachstes System associirter Formen "
{vkU Biruiren Far)nen,p. 330) and its Generalization.
Let (a, 6, c, . . . . ^, ? ][ x^ yY be any binary quantic. Let the provector
symbol (M* + 2^:^^ + ^hhg + ....) be denoted by ft, and the revector symbol
{ahb + 26ie + 3c5rf +....) by U. Let ^2^ represent the quadrin variant of
the above form when n ■=, 2i. Now let ft and U be made to comprise the
2i + 1 letters a, 6, c, ....?, m; then aHQoi — 26^2* will be nullified by the
operation of U and will therefore be a cubin variant for the case of n = 2e + 1
which we niay call Q^i ,. i. Also lef Qo = « ; then Qo? Qd Q25 • • • • Q^ ^^^^ ^®
diflferentiants to all binary quantics of degree equal to or greater than (i.
The above I call basic differentiants. Their distinguishing characteristic is
that the highest letter in each of them enters into it only in the first degree
multiplied by a or by a^ and by no other letter. Now let I) be any given
diflferentiant of degree (i and for the moment make a = 1. Then it is obvious
that J) may be expressed — by means of successive substitutions of its ultimate,
its penultimate, its antepenultimate, etc. letters up to c inclusive, in terms of the
corresponding basic diflferentiants and the anterior letters, — as a rational
integer function of Qi, Qz^ . . . . Q^, b] or, restoring to a its general value, will
be a rational integer function of Qo? Qu Q25 • • • • Qm? ^> ^^J ^> divided by a
power of a. But I say that b will have disappeared in the process. For
UZ) zz ; and UQo = 0, 15 Q, = 13 Q^ = 0. Hence, regarding each Q
as a constant, (a-—)F=0^ or i^does not contain b.
*For by a well known formula if /) is a differentiant in x of the type [10 : i,j\^ f5QD = (t; — 2w) D. Con-
sequently when Qm is regarded as a differentiant in x of the type [2t: 2i -f h 2] ^^Qm = Qii Also OQ^ =
and 06 = o. Hence (oCQ^ — 26 Q«) = 0.
Sylvester, On ClehscKs Theory of the ^^ Einfachstes System,^^ (tc. 119
Again, suppose we take a system of two quantics and let Qoj Qd • • • • Cm ^
the basic differentiants of the one, Q^, Qi, - - * * Qv of the other, and let 2) be
any differentiant of the system. Then by the same method as before we
shall find
Also each Q will be nullified by U, and each Q by U^, and therefore each Q
and Q as well as I) will be nullified by the operator U + U^. Hence
we shall have
\ db^ db'J
or
F=q){al/ — db) ,
^ being a rational integral form of function. In like manner for a system of
three quantics, regarding the several sets of its basic differentiants as con-
stant, we shall have
F=q>{al^ — a!b\ acf — a!c: bd — Vc) ,
where ^ is a rational integral form of function, or
F zz.'^ {ah — o[b\ (uf — a!c: a, of) ,
and so in general. Hence, remembering that any relation between differen-
tiants must continue to subsist between the co variants of which they are the
roots, and now, understanding by base forms the complete covariants of which
the basic coefficients are the roots, we may pass from differentiants to in- or
CO- variants and obtain the following theorems.
1°. For a single quantic of degree i, any in- or co-variant is expressible
by a fraction whose numerator is a rational integer function of its i base
forms and whose denominator is a power of the quantic. This is Clebsch's
theorem.
2°. For a system of quantics, any in- or co- variant is expressible by a
fraction whose numerator is a rational integer function of the separate base
forms of its several quantics and of any complete system of (ji — 1) inde-
pendent Jacobians of the quantics taken in pairs, and whose denominator is a
product of powers of the quantics of the system.
Also it will be observed that these theorems will continue to subsist when
the base forms have for their roots in lieu of the basic differentiants, as above
120 Sylvester, On ClebscKs Theory of the ^^Einf achates System^'' (kc.
defined, any ascending scale of diflferentiants in which the letters enter suc-
cessively one at a time and each letter on its first appearance figures only in
the first degree and combined exclusively with powers of a.
On the theory of basic forms may be grounded a method for obtaining,
in propria persona^ the fundamental in- and co- variants to a quantic or system
of quantics in regular succession, by a process which continues so long as
there are many more to be elicited and comes to a self-manifesting end as
soon as the last irreducible form has been obtained, like an air pump that
refuses to act as soon as the exhaustion has become complete. In a word, the
cataloguing of the irreducible in- and co- variants is transferred to the prov-
ince of, and becomes a problem in, ordinary algebra.
I have previously observed that any expression which represents a
diflferentiant in regard to a quantic of a given degree necessarily does the
same for quantics of all higher degrees. And I may take this occasion to
remark, or to repeat, that a differentiant may be irreducible in respect to the
quantic of minimum degree to which it can be referred, and yet not so for
quantics of higher degrees. Thus, if we take the expression
a^d" + 4ar^ + U¥ — 36 V — Qahcd ,
this referred to a cubic is irreducible (as is well known), but regarded as a
diflferentiant of a quartic or higher degreed quantic, is reducible, being in
fact identical with
{ac — V){ae — Ud + 3c^) — a
Let us suppose a linear function yu — xv combined with a quantic into a
system. Then it follows as a corollary from (2° at p. 118), that if the quantic
belongs to the form (a, 6, c, . . . . Z ][ u^ v)% or say more simply to the form
[a, 6, c, .... Z] any co variant of such quantic multiplied by a suitable power
of a will be a function of y, ax + hy and of the diflferentiants, or in a word,
every covariant of the quantic expressed as a function of x and as -^-hy will
have no coeflBcients but what are diflferentiants, or to use Professor Cayley's
term, semi-invariants. Thus, ex. gr., the Hessian of the cubic (a, 6, c, d\x^yY
may be put under the form
-^2 { (^ — *') (^^ + ^y+ (^"^ — 3«»^ + 2fi') {ax + hy)y-^{ac — ¥yfY
a, 6,
c
i, c,
d
c, d,
e
Sylvester, On ClehscKs Theory of the ^^Einfachstes System^^^ &c. 121
So it will be found that the Hessian of the quintic, viz.
{ae — Uc + Zc')x' + {af— She + 2cd) xy + {bf— 4cd + 3d^) f
on writing ax -^^ hy ■=. X becomes
}iA{ae — 4bc + Sc") X' + {ay— babe + 2acd + SbH — Qbc") Xy
_ [[ac — 6^) {ae — 4bd + Sc") + 3a {ace + 2bcd — ad^ — b^e — (^)]f\ ^
where all the coeflBcients are semi-invariants-in-a:, the second coeflBcient being
one of the basic differentiants and the latter part of the third coeflScient, the
catalecticant
a, ft, c
6, c, d
c, rf, e
and so more generally, it may be shown to follow from (2°), that if there be
any number of binary quantics
[a, 6, c . . . .] , [a', 6', c/, . . . .] , [(fy 6^, c^ . . . .] ,
every covariant of such system, expressed as a function of y atid of any one of
the quantics
ax + by ^ afx -}- b'y J
chosen at willj has differentiants-in-x exclusively for its coeflScients.
It is easy to express the base-covariants in terms of the roots. Those of
weight 2n and order 2 will be of the form
where i^ may be expressed as
(«1 — (hYi^Z — (liY («2n-l — «2n)' ,
or, {dl-^ «2)(«2 ^){(h ^4) («2n-l «2»)(^n «l) J
or under a variety of other forms all equal to a numerical factor pres ; for the
type [2n : 2/1, 2] and the more general one [2n : 2n + v^ 2] are monadelphic.
And again those of the weight 2n + 1 and order 3 may take, or at all events
be replaced by, the form
, ^8 \ S
2t yOi — 0^2 • ^ a^* •• ^n— 1 ^n • ^n — ^1 • ^1 ~ ^n-f 1 • ^ — ^1 • *^ — ^ • • • •^ ^n+1 • *^ ^n+2 • ^ ^in+8 • • •/
It is proper to notice that the type \2n + 1 : 2/1 + 1 + r ; 3] is only monadel-
phic so long as 2;i + 1 is less than 9, so that we cannot, without an investiga-
tion which might be tedious, determine whether the above representation
coincides with the basic forms of the third order in the coeflScients adopted
in p. 118 ; but such investigation would be a work of supererogation, for the
only material character for any of the base-covariants in question to possess
81
122 Sylvester, On ClehscKs Theory of the ^'Einfachstes System^^^ Ac.
is, that its root differentiant-in-j: shall be not higher than of the third order
in the coeflScients and shall contain the element egn + 1- Any formula having
this property (which is enjoyed by the root function above given) is just as
good as any other for the purposes of this theory.*
It will be seen to follow from the theorem I have given for diflferentiants
from which Clebsch's follows as an immediate consequence, that all the per-
mutation-sums of any rational integer function of .the differences of the roots
of an algebraical equation of the nth degree are rational integer functions of
{n — 1) of them of the second and third order alternately ; so, for example,
all the coeflBcients in Lagrange's equations to the squares of the differences of
the roots of an algebraical equation in its ordinary form are rational integer
functions of {n — 1) known quantities. Thus, for instance, the equation to
the squares of the differences of a cubic equation will be
p3 + 18 (6^ — ac)f + 81 {V — acf + 27A = ,
where the coefficients are given in terms of two differentiants (6^ — ac) and A.
Throughout this paper the perspicuity of expression has been consider-
ably marred by want of a complete nomenclature which the theory of graphs
and types necessarily calls for and which I shall hereafter employ whenever
I may have occasion to revert to the subject. It is as follows :
In the first place, w, the weight in respect to the selected variable, and j,
the order in the coefficients, are terms well understood and need no change or
further illustration ; i, the degree of the parent quantic, I shall hereafter call
the rank of the type, ij — 2w which becomes the degree of a covariant got by
expanding the differentiant of type [w : i, j] may be called the grade. The
order and rank may be termed collectively the permutable indices.
When a differentiant is given algebraically its weight and order are
given but not its rank ; in addition to the weight and order a third number
which may be called the range (and which I shall denote by a Greek e) is
• Writing the type under the form \2n + 1 : 2n + 1 -|- v, 8]. the degree of the corresponding covariant
in the variables is 2n -|- 1 -|- 3v, which is the degree in x of the symmetrical function assumed in the text ; also
each letter in this function occurs 8 times agreeing with the order 8 of the type, and the number of factors in
the coefficient of the highest power of x is 2n -|- 1, which is right for the weight. It is obvious also by inspec-
tion that the product a^.a^, . . . ^2114- 1 ^^^^ m\&q frcm each term of the assumed symbolical function affected
always with the same sign, so that Cj* ^ 1 will occur (as required) in its expression in terms of the coefficients.
Of course all the same conclusions will apply if in the formula (Oj — a,) 2 (a, — 0^)2 .... (a^^__^ — <»an)2 is
substituted in lieu of (a^ — a^) [a^ — a,) .... {a^n — i — »2*»)(^a» — ^\)'
That the type to which Q^n + 1 belongs is non-monadelphic from and after 2n + 1 = 9 is obvious from the
fhd that that type, when the degree of the parent quantic is made a minimum, is of the form [2n -^ \i 2n
-\- 1, 8], the multiplicity of which is the same as that of [2n + 1 : 3, 2n+ 1], or set out in full [2n+ 1 : 8, 2n
-j- 1 : 2n -|- I] i ^ut cubics include covariants of orders and degrees 2 : 2 and 8 : 8 among their fundamental
forms, and 9 : 9 can be formed either by taking a triplication of 8 : 8, or by combining 8 : 8 with a triplication of
2 : 2, so that when 2fi -j' ^ ^ ^ ^^® ^7P® ^ diadelphic, and a fortiori^ it is non-monadelphic for values of 2n -4- 1
superior to 9.
Sylvester, On CUhscKs Theory of the '' Einfachstes System,'' &c. 123
given, being the number less 1 of the letters which enter into it. The
relation between rank and range is one of inequality. The former may be
equal to, or greater than, but not less than the latter.
The multiplicity of the type to which a given'diflferentiant belongs is a
function of the weight, order and rank and is consequently not known until the
rank is assigned. Thus, ex. gr. {ac — ft^)^, considered as having the lowest
possible rank, viz. 2, (the range) is monadelphic ; its type is then [2 : 2, 4], but
if the rank 4 be assigned to it so that its type is [2: 4, 4], it becomes diadel-
phic. We have then, in general, 6 characters (not all independent) apper-
taining to a differentiant, viz., weight, rank, order, grade, range and multiplicity.
The theory of types has never hitherto formed the subject of distinct contem-
plation, and that is why the necessity for the use. of some of the above terms
has not been previously felt. But it will have been observed that throughout
the preceding memoir it has forced itself upon our notice, and in particular,
that it is impossible to go to the bottom of the so-called law of reciprocity or
that of the radical representation of forms without keeping in view the
question of type and multiplicity.
I have also to remark that since the preceding matter was completed I
have been surprised to learn that recent chemical research favors the notion
of simple elements (hydrogen atoms in special) being distinguishable from each
other in chemical composition. If this view is confirmed, the discrepancy,
which I have pointed to, between the known conditions for the existence of
algebraical graphs and the unknown natural laws which govern the produc-
tion of chemical substances may become partially or wholly obliterated, so
that, for example, the hydrogen molecule and the extended derivatives from
marsh gas may exist in accordance with, and not in contradiction to, alge-
braical law, and thus it is possible to conceive that all the phenomena of
chemistry and algebra may ultimately be shown to be identical.
Since the above matter was sent to press I have been led to study alge-
braically what may be termed the direct problem of isomerism, that is to say
the determination of the number of combinations subject to given conditions
that can be formed between the constituents of groups each containing a given
number of equivalent chemical atoms, the valences of the several groups being
either independent or given linear functions of a certain number of indepen-
dent parameters. In this problem the numbers of atoms are given and the
valences left indeterminate. In the inverse problem the valences are given
and the numbers left indeterminate.
124 Sylvester, On ClebscVs Theory of the '^ Einfachstes System,^' drc.
The problem of the enumeration of the saturated hydro-carbons, investi-
gated by Professor Cayley, is a simple example of the inverse problem. The
direct problem admits of a uniform and unfailing method of solution by gene-
rating functions, the exposition of which may probably form the subject of an
additional Appendix. in the following number * This method is substantially
the same as that which I have described in general terms in the Ckmtptes jRen^
dus as applicable to the theory of ternary and other higher varieties of quan-
tics but less diflScult of application to the Isomeric Problem on account of the
greater simplicity of the crude forms f subject to reduction, which appear in it.
Appendix 4 will contain the application of the theory of " Associirter Formen "
to the algebraical deduction of the irreducible forms of the quintic and certain
other cases which but for the press of matter awaiting publication in the
Journal would have formed part (as announced) of the present Appendix.
As already stated in a previous foot-note, the theory of irreducible forms
reappears in the isomeric investigation, the general character of the reduced
generating function to be interpreted in it being precisely the same as in the
invariantive theory, which constitutes an additional and a closer and more
real bond of connexion between the chemical and algebraical theories than
any which I had in view when I commenced the subject of this memoir.
* The principle employed in this method leads to the following theorem only a particular case of which
comes into play in the general partition problem which covers the ground occupied by the allied invariantive
and isomeric theories. Let there be given a product of a limited number of rational functions of
Ui^.U2^. . , . Ui *Ui \ 1*2 ... . Ui •; etc., etc.,
where all the indices are positive or negative integers, and let /ii, //ji • • • /'2 ^ given linear functions of
Vj. V21 . . . vj (j being not greater than i), then it is always possible to find a limited product of rational
functions of
v^^ .v^. . . . v^] rj'* . tf * . . . . t^> ; etc., etc.,
where the indices are exclusively /)o«i<tt?«, such that the coefficient of t>][^ . vJ* . . . t^' , in their product developed
according to ascending powers of ©j, ©ji . . . »> , shall be the same as the coefficient of Uf* Mj*. . . . u^ in the
original product developed according to ascending powers of u,, w,, . . . . u, . Previous to the discovery of
this principle the problem of isomerism, now completely solved potentially for the direct case, must have remained
unattackable by any existing methods, such for example as were known to Kuler, the inventor of the applica-
tion of the method of generating functions to the theory of partitions. It renders supererogatory a large part
of the methods devised by myself for the treatment of the problem of compound partitions contained in the
printed notes of my lectures on Partitions, delivered at King's College, London, in the year 1859. As an
example of the direct problem of isomerism, suppose that three atoms of the same valence j are to combine with
F atoms of hydrogen which do not combine inter ae; then the number of combinations which can be so formed
1 -1. fljf _L ^23.2
is the coefficient of ofx* in the development of the fi:eneratinfi: function .- , rr-; -, if the
'^ & & (l — o»)(l —.aar)«(l —aar»)
three atoms are all unlike, and of the venerating function — — — , .. , — if they are all
* * (1 — a»)(l — oa:)(l— o2a:a)(l — aa?») ^
alike
f Very unluckily printed as ^^ formes eubiqttes " in the Comptea Rendus.
NOTE ON THE LADENBURG CARBON-GRAPH.
The reasoning by which I have established, in the preceding number of the Journal, the
validity of the Ladenbnrg graph (and the invalidity of the Keknlean one) as a representative of the
root differentiant to a co^ariant of the 6th degree in the variables and of the 6th order in the coeffi-
cients to a qnartic/ is so peculiar and it may seem to some of my readers so far-fetched, that it
appears highly desirable to confirm it by a direct demonstration founded on the principle, that the
permutation- sum of the product of the bonds in a vnlid graph interpreted as differences between the
letters which they connect, shall not vanish. Previous to applying this principle to Ladeuburg's
graph we must convert it into an invariant by attaching hydrogen atoms to the six apices. Let these
apices be called a, 6, c, rf, c, /, and the hydrogen atoms a, /3, y, <J, e, ^ : then the permutation-sum
under consideration is
2(a-6)(a-c)(6-c)(rf-e)(rf-/)(c-/)(a-^)(6-«)(«-/)(a-a)(6-/?)(c-y)(rf-d)(«-e)(/-0)
where the 6 letters a, 6, c, rf, e, / are intcrpermutable, as are also the 6 letters a, /3, 7, J, e, ^.
It may be well to observe at this point that if we struck off the hydrogen atoms and treated the
graph as representing an invariant to a cubic form, the permutation-sum
2 (a — 6) (a - c) (6 — c)^ [d — e) {d — /) (e -/)(« — d) (d — c) {c — /)
would be found to vanish, as may easily be shown and as it ought to do, because there exists no
invariant of the 6th order in the coefficients to a cubic form. Let a and d be interchanged in the
term given under the sign of summation in the permutation-sum formed from the Ladenburg
graph; then the sum of this together with the original term becomes
(a-d) (6-«) (e-f) {b-e) (e-f) (6-/3) (c-y) (^-e) (/- 0)
multiplied by
(a<J — rfa)(a2— A+Ta + ftc) (d^ ^ f+f d + ef) —(d6 — aa) (d^ — b + c d+be) {a^ ^^ff a + ef),
which last named multiplier will be 'found to contain the quantity (a^d^ — a^d^) {a + 6). Again, in
the multiplicand, let b and c be interchanged ; then, since
(6 _- e) {c -/) — (c — e) (6 — /) = (6 — c) {e -/),
the sum of the original and permuted multiplicand will contain a term
(a - rf) (6 — ey (e — /)» be (e — (/ - 1>h
and accordingly the entire permutation-sum will contain the terms
The partial sum last written is
Hence we may readily see that the total permutation sum will contain inter alia a positive multiple
of the combination a*b*e^d^cfa and will not vanish, and consequently the graph is valid and not
illusory ; 1 presume that the same method applied to Kekule's graph regarded as a representation of
the covariant to the type [9 : 4, 6 : 6], which is the same thing (except that the hydrogen atoms
are suppressed) as the graph to the invariant [15 : 4, 6 ; 1, 6; 0], would serve to show it to be illu-
sory as previously inferred from other considerations.
EBRATA IN THE PART OF THIS MEMOIR INCLUDED IN THE PRECEDING NUMBER OF THE JOURNAL, pp. 64-104.
Page 66. lIne4,/orqaarticrradquadric. p q. ii-jp o for—- l-(aB read -^ J-/
Page 66, Une 4, (from foot) /or irrefragible read irrefragable. ^^ ' ' '' 16^^ 216 (*^-
Page 70, line 17, q/3r«r covariant tnjiw/ of. Page 97, line 5, (from foot,) /or a A: fi: »» read aa A: <*: v.
Page 74, line6,/or the aecond read Fig. 15. Page 98, line 9, for /a 1^2 = afUi + b'l>2 + CUs read Hl/i.
Page 74, lines 25, 29, /or Fig. 43 read Fig. 45. = aUi -f bUi 1 cU^.
Page 76, line 11, /or pluralty read plurality. Page 98, line 18, for which makes read which make.
Page 77, line 5, (from foot), /or CA read CB. Page 99, line 5, /or none read no.
Pago 79, last paragraph, /or x read y. Page 99, line 19, /or XU + in V read XU + M K
Page 80, line 10, /or a 2d read a 2 J. Page 101, line 6, dele period at end of line.
Page 86, line 1&,for enparametrlc read henparametric. Pi^e 101, line 20, after biologists intert term.
Page 87, line 12,./or expressions read expression. Page 101, line 3, (from foot,) for yr read jr.
Page 91, line 15, for j read i. Page 102, middle of page, all the indices of c should be tubseript.
Page 93, line 12, for abe read ace. VAgo 102, foot-note, /or 9 — 1 read q-^i.
Page 93, line 16,/or maltiplier read multiple. Page 1(^ last line of text,/or concluaiou read conclusion.
32 125
k
EXTRACT OF A LETTER TO MR. SYLVESTER FROM PROF.
CLIFFORD OF UNIVERSITY COLLEGE, LONDON.
The subjoined matter is so exceedingly interesting and throws such a
flood of light on the chennico-algebraical theory, that I have been unable to
resist the temptation to insert it in the Journal^ without waiting to obtain the
writer's permission to do so, for which there is not time available between the
date of its receipt and my proximate departure for Europe. It is written
from Gibraltar, whither Professor CliflFord has been ordered to recruit his
health, a treasure which he ought to feel bound to guard as a sacred trust
for the benefit of the whole mathematical world. J. J. S.
" The new Journal I look forward to with the greatest interest : it will
be the only English periodical in which one will have room tjo print formulae,
except the Philosophical Transactions. I had designed for you a series of
papers on the application of Grassmann's methods, but there is only one of
them fit for printing yet. It is an explanation of the laws of quaternions and
of my biquaternions by resolving the units into factors having simpler laws
of multiplication ; a determination of the corresponding systems for space of
any number of dimensions; and a proof that the resulting algebra is a
compound (in Peirce's sense) of quaternion algebras. It thus appears that
quaternions are the last word of geometry in regard to complex algebras.
"Another of them was to be about the very thing you speak of, which
was communicated to the British Association at Bristol, not Bradford. There
is no question of reclamation, because the whole thing is really no more than a
translation into other language of your own theories published ages ago in
the Cambridge Mathematical Journal.* I have a strong impression that you
* Not having access to the reports of the British Association, I wrote to Professor Cliffbrd to the effect that
I had an impression that he bad made a communication to the meeting at Bradford which had some affinity to
my chemico-algebraical theory and requesting him, in that case, to claim back what was his own.
I think he overstates the obligations which he alleges to my previous papers. At all events he has more
than reconquered his title to the merit of the first conception by the completeness he has imparted to it, for
whilst I was only able, in certain cases, to represent in terms of the roots of the parent quantic, the quantita-
tive constitution of a form pictured by a graph, through the instrumentality of the reciprocal figure, he has
given a direct rule for finding in all cases the algebraical content of a graph in terms of the coefficients, from
the graph itself. In a word he has found the universal pass key to the guantijication of graphs. It seems to me
ID the highest degree improbable that our joint speculations should not eventually find their embodiment in chemi-
cal doctrine proper, and I think that young chemists desirous to raise their science to its proper rank would act
wisely in making themselves masters betimes, of the theory of algebraical forms. What mechanics is to physics,
that I think algebraical morphology, founded at option on the theory of partitions or ideal elements, or both,
is destined to be to the chemistry of the future. Brodie, in his Chemical Calculus, seems to have haH an instinc-
tive appreciation of this truth. I have previously called attention to the fact that invariants and isomerism
are sister theories.
126
Extract of a Letter to Mr. Sylvester from Professor Clifford. 127
will find there the analogy of covariants and invariants to compound radicals
and saturated molecules.*
*'I consider forms which are linear in a certain number of sets of k
variables each. To fix the ideas, suppose A: zz 2 and that I have altogether
6 sets of 2 variables each, namely
Suppose the forms are
{xyzu) , [yzvw) , {xv) , [uw) ;
viz. {xyzu) means an expression separately linear and homogeneous in the x^
the y, the z^ and the w, and so for the rest. I observe that in these four forms
each set of variables occurs twice. This being so, there is one invariant of
the four forms, which is invariant in regard to independent transformations of
the six sets of variables. This you knew thirty years ago. All I add is : f
to obtain this invariant^ regard the variables as alternate numbers^ and simply
multiply all the forms together. By alternate numbers J I mean those whose
multiplication is polar {xy = — yx) and whose squares are zero. The pro-
duct of the forms will then be eqtial to the invariant in question mul-
tiplied by the product of. all the variables. The quartic forms may be
represented by the symbol -o- , the quadratics by -o- . Thus the invariant
(xyzu) {yzvw)(xv) {uw) will be represented by the figure | /^| ; whereas, (xyzu)
0-^0
(yzvw) (xu) (vw) is this form 0-0^0^=0. The former is clearly the product
of the two quartic covariants _^ ^_ got by cutting it across the dotted
lines; while the latter is the product of the quadrico variants 0=^0^,
=0- o . A bond between two forms means a set of variables common to
* 1 feel induced to say of Professor Clifford what a great master in another field is reported to have done
of a certain famous ** Oxford graduate/' "That young man sees many things in my works which I was not
before aware was contained in them," and to regard his generous confessions of obligation as the graceful homage
of a young athlete to a half-spent veteran in the arena. What I have seen as in a glass darkly by the uncertain
light of analogy, he has viewed by direct vision and brought out into the full blaze of day.
t " All that Professor Clifford adds " is the very pith and marrow of the matter which before was wanting.
J The term alternate numbers (Hankel's, if my memory serves me right) applied to letters, subject (like
Hamilton's t,j, A,) to the law of polar multiplication, seems to me very inadvisable to introduce into the subject.
There is no question of quantity, still less of number; and the epithet alternate has no meaning except by oblique
reference to alternants in the sense of determinants. I think the right term is that first employed by myself
and which 1 shall cortainly continue to employ undeviatingly, viz. '* polar elements." In like manner I protest
against my most expressive and suggestive word '* cuniulants " being ignored by Mr. Muir and replaced by the
unmeaning and ill chosen word " continuants." J. J. S.
128 Extract of a Letter to Mr. Syhester from Professor Clifford.
them. Of course, we may regard two or more of the forms as identical, and
O-O .
so form invariants of a single form; thus | is the discriminant of a
cubic* Of course, the main thing is to pass from this system of
separate variables to that in which the same variables occur to higher orders
in the same form, or back again — what you call ' unravelment.'
'' The part of the theory which astonished me most is its application to
intergredient variables wiijen the number in a set is greater than 3, — such
as the six co-ordinates of a line in the case of quaternary forms. When the
original variables are^ regarded as alternate numbers, these intergredients are
simply their binary products. Thus by simply multiplying the linear forms
representing two planes, we get an intergredient form representing their line
of intersection. And so generally, whatever be the number of variables in a
set, the intergredient variables are merely their products so many together.
With this understanding, the product of a set of forms in which the variables
are regarded as alternate numbers is the only invariant or covariant of the
forms which possess certain definite characters of invariance.
" The ordinary theory of symmetrical forms seems to me to bear the
same relation to this one (of forms linear in several sets of variables) that
a boulder does to a crystal — all the angles rounded off so that you can't
see through it so clearly "
* I will take the example of this figure to illustrate Professor Clifford's rule for finding the algebraical eoft-
X t
tent of a graph. Let the bonds be called ,. ^ . v* Then there will be 4 forms corresponding to the 4 apices or
atoms, yiz.
(«n ^21 ^ii «4» <'5» ^B> «7» «») (^if ^z) (yi» Vi) (*i» «a)f
(*1» *2» *8» *4» *5» *6» *7I *») (^n ^l) (^1» ^j) («*!» "j)»
(Ci, Cj, Cj, c^, C5, Cg, Cy, Cg) (<i, <j) (Wi,Uj) («!, r,),
where all the .r, y, z, i, m, v letters are to be regarded as polar elements. Take the polar product of these forms ;
the coefficient of
*^l * *^2 * ^1 * ^2 * ^1 * ^2* ^1 • ^2 * ^1 * ^2 * ^1 * ^2
will be an invariant of 8 linco-lineo-linear forms.
If we make the values identical for the same index, whatever the letter which it affects, it becomes an
invariant of a single lineo-lineo-linear form ; and finally, if we make the coefficients of x^y^z^^ y^z^ar,, z^rr^y, all
alike, and again the coefficients of x^^Zy^ Vi^t^n ^i^zVi ^^^ alike, and identify the letters x^y^z^ the form
becomes a binary cubic ai:d the invariant becomes its discriminant, [^^uaere^ whether this beautiful use of the
method of polar multiplication is not, in its ultimate essence, identical with Professor Cayley's original method
of hyperdcteiminants.] We know a priori by my permutation-sum test that the algebraical content above
indicated will not vanish because 2 (a — 6; * (a — d)^ (a — c) {b — d) is not zero, whereas the algebraical content
of the figure formed by turning round one of each pair of the doubled lines into the position of the two diagonals
respectively will vanish because the permutation-sum of Z (a — 6) (6 — c) (c — d) (d — a) {a — e) {b — rf) is zero.
J. J. S.
\
RESEARCHES IN THE LUNAR THEORY.
By G. W. Hill, Nyack Turnpike, N. Y.
(Continued from p. 26.)
Chapter II.
Determination of the inequalities which depend only on the ratio of the mean motions
of the sun and moon.
If the path of a body, whose motion satisfies the diflferential equations
intersect the axis of x at right angles, the circumstances of motion, before
and after the intersection, are identical, but in reverse order with respect to
the time. That is, if t be counted from the epoch when the body is on the
axis of or, we shall have
X = function [t^) , y =- t - function {t^) .
For if, in the differential equations, the signs of y and t are reversed, but that
of X left unchanged, the equations are the same as at first.
A similar thing is true if the path intersect the axis of y at right angles ;
for if the signs of x and t are reversed, while that of y is not altered, the
equations undergo no change.
Now it is evident that the body may start from a given point on, and at
right angles to, the axis of a:, with different velocities; and that, within
certain limits, it may reach the axis of y, and cross the same at correspond-
ingly different angles. If the right angle lie between some of these, we
judge, from the principle of continuity, that there is some intermediate
velocity with which the body would arrive at and cross the axis of y at right
angles.
129
130 Hill, Researches in the Lunar Theory.
The difficulty of this question does not permit its being treated by a
literal analysis; but the tracing of the path of the body, in numerous
special cases, by the application of mechanical quadratures to the diflferential
equations, enables us to state the following circumstances : —
If the body be projected at right angles to, and from a point on, the
the axis of x, whose distance from the origin is less than 0.33 . . . \^ , there
is at least one (near the limit there are. two) value of the initial velocity, with
which the body, in arriving at the axis of y, will cross it at right angles.
Beyond this limit it appears no initial velocity will serve to make the body
reach the axis oi y under the stated condition.
If the body move from one axis to the other and cross both of them per-
pendicularly, it is plain, from the preceding developments, that its orbit will
be a closed curve symmetrical with respect to both axes. Thus is obtained a
particular solution of the diflferential equations. While the general integrals
involve four arbitrary constants, this solution, it is plain, has but two, which
may be taken to be the distance from the origin at which the body crosses the
axis of X and the time of crossing.
Certain considerations, connected with the employment of Fourier's
Theorem and the po?sibility of developing functions in infinite series of
periodic terms, show that, in this solution, the co-ordinates of the body can
be represented, in a convergent manner, by series of the following form,
X = Ao cos Iv {t — 4)] + ^1 cos 3lv{t — #o)] + ^2 cos 5 [y {t — 4)] + • . . ,
^ = ^osin [v {t—to)] + B,sm3\v {t— to)] + B.^in 5[v (^— f,)] + . . .,
where <„ denotes the time the body crosses the axis of or, and — is the time
V
of a complete revolution of the body about the origin. We may regard v
and to as the arbitrary constants introduced by integration ; the coeflBcients
Ao, Ai . . . B^^ Bq . . . are functions of ^i^ n' and v.
For convenience sake we may put
-4f zz a^ + ^— i-i » Bi ziz B,i — ft— <— 1 •
Then, r being put for v{t — 4)j the series, given above, may be written
X = Xi.SLi cos {2i + 1) T,
j^ = 2,-.a, sin {2i + l)r,
the summation being extended to all integral values positive and negative,
zero included, for i. By adopting polar co-ordinates such that
a: = r cos 4) , y =i r sin 4) ,
Hill, Researches in the Lunar Theory. 131
and writing v for 4) — r, that is, for the excess of the true over the mean
longitude of the moon, the last equations are equivalent to
r cos u = 2,-. a, cos 2iT,
r sin i; = 2,. a^ sin 2ir,
In order to avoid the multiplication of series of sines and cosines, and
reduce evarything to an algebraic form, for x and y, we substitute the imagi-
nary variables u and- 5, and put ^ = f' ^"^ We have then
W Z^i» a^^ , 5 ^i* ^ t Vy •
f will always be employed as the independent variable in place of t or r.
Denoting the operation ^ —•:= — V — 1— by the symbol 2), so that, in
general,
D «0 = KS
and taking the liberty of separating this symbol as if it were a multiplier,
and moreover putting
W n' u
m = — = , X = ~ ,
V n — n' V
the diflferential equations, determining u and 5, given in the preceding chap-
ter, may be written
It will be noticed that either of these equations can be derived from the.
other by interchanging it and s and reversing the sign of m or D. We may
also remind the reader that they determine rigorously all the parts of the
lunar coordinates which depend only on the ratio of the mean motions of the
sun and moon and on the lunar eccentricity. The Jacobian integral, in the
present notation, is
Du.Ds^- -^ +-^m^(i^ + sV= C.
{us)i 4
The most ready method of getting the values of the coefficients a,, is that
of undetermined coefficients; the values of u and s, expressed by the pre-
ceding summations with reference to i, being substituted in the differential
equations, the resulting coefficient of each power of ^, in the left members, is
equated to zero, which furnishes a series of equations of condition sufficient
132 Hill, Researches in the Lunar Theory.
to determine all the quantities a<. For this purpose we may evidently employ
any two independent combinations of the three equations last written, and
it will be advisable to form these combinations in such a manner that the
process of deriving the equations of condition may be facilitated in the
largest degree. Now it will be recognized that the presence of the term
' — - , in one of the factors of the differential equations, is a hindrance to
their ready integration, being the single thing which prevents them from
being linear with constant coefficients. Hence we avail ourselves of the
possibility of eliminating it. Multiplying the first differential equation by 5,
and the second by u, and taking, in succession, the sum and difference,
uB's + slfu—2m (uDs — slhi) — -^— + ~ m^ (u + sY = 0,
(us)t 2
um — sjfu — 2m {uDs + sihi) +%^^ (^' — ^) = 0,
then, adding to the first of these the integral equation, and retaining the
second as it is, we have, as the final differential equations to be employed,
Lf" {us) — Du.Bs —2m {uDs — sDu) + ^ m^ (w + sf = C,
J) {ul)s — sDu — 2mus) + -|- ^' (^' — ^) =^'
It must be pointed out, however, that these equations are not, in all
respects, a complete substitute for the original equations. It will be seen that
\L or X, an essential element in the problem, has disappeared from them, and
that, in integration, an arbitrary constant, in excess of those admissible, will
present itself. This will be eliminated by substituting the integrals found in
one of the original differential equations, in which /t^ or x is present ; the
result being an equation of condition by which the superfluous constant can
be expressed in terms of ^ and the remaining constants.
We remark that the left members of our differential equations are
homogeneous and of two dimensions with respect to u and s. If the first
were differentiated, the constant C would disappear, and both equations would
be homogeneous in all their terms. This property renders them exceedingly
useful when equations of condition are to be obtained between the coefficients
of the different periodic terms of the lunar co-ordinates, and it is for this
purpose that we have given them their present form.
Hill, Researches in the Lunar Theory. 133
From the signification of the symbol D,
Du:=il.,, {2i + 1) a,^2'^ \ Ds = ^,,{2i + 1) a_,_,^ + * , '
jy^u = 2,. (2i+ l)2a,^2' ^S U^s zzX-. (2^ + l)'a_,_,^« + * ;
also
• t^zz2^..[2,.a,a,_J^2^
u" = Ij . [2, . a,a_, ^ ^._ J ^'•^' ,
s^ =^j. [2, . a,a_,_,._i] ^'> ,
])u.I)s = —Xj, [2, . (2/ + 1) (2/ - 2j + l).a,a,_,] ^^- ,
uBs — sDu = — 22y . [2, . (2/ — j + 1) a,a,^J ^^-^^ ,
where the summations with reference to j have the same extension as those
with reference to i. On substituting these expressions in the differential
equations, and equating the general coefficients of ^^^ to zero, we get
2,. [{2i + l)(2i - 2j + 1) + 4/ + 4{2i-j + 1) m + |-m^] a,a,_,
+ — m2 2,. [a^a_.^y_i + a,a_,_y_ij =0,
4; 2,. [2i— j + 1 +mja,a,_^ — — m'2i. [a,a^<+y^, — a,a_<_^_ij =0,
which hold for all integral values of j both positive and negative except that,
when j = 0, the right member of the first equation is C instead of ; but as
the second equation is an identity for j =: 0, for the present this value of j
will be excluded from consideration.
By multiplying the first equation by 2, and the second by 3, and taking
in succession the difference and sum, the simpler forms are obtained,
2,. [8i^ — 8 (4j — 1)2 + 20/^ — 16j + 2 + 4 (4i _ 5j + 2) m + 9m^ a^.^
+ 9m^2-.aa =0
2, . [Si^ + 8 (2j + 1) i - 4/ + 8j + 2 + 4 (4^ + j + 2) m + Om^ a. a, _^
+ 9m^ 2i. a<a_i_y_i = 0.
These two equations are not distinct from each other, when negative, as well
as positive values, are attributed to j. For if, in the expression under the first
sigh of summation in the first equation, we substitute, which is allowable, for
t, i — j, and — j for j throughout the equation, the result is identical with the
second equation. This is explained by the fact that we get all the inde-
pendent equations of condition, these equations are capable of furnishing, by
attributing only positive values to j. Hence, allowing j to receive positive
34
134 Hill, Researches in ike Lunar Theory.
and negative values, all the equations of condition can be represented by a
unique formula.
Although the number of these equations is infinite, and also that of the
coefficients a^ , it is not difficult to see that the first ought to be regarded as
one less than the second; and that, in consequence of the bi-4imensional
character of the equations, they suffice to determine the ratio of any two of
the quantities a^ in terms of m. It will be seen, from developments to be
given shortly, that if m is regarded as a small quantity of the first order, a^ is
of the ± 2i^ order. It will be advisable then to select ao as the coefficient to
which to refer all the rest ; and we shall have, in general,
a^ = ao -F (m) .
The equations of condition, as written above, determine the a< in pairs ; that
is, if we put J = 1, we have the equations suitable for determining ai and a_i,
and, in general, the equations, as written, determine a^ and a_y. And, as
they involve both these quantities, it will be advantageous to eliminate
approximately each in succession, as far as that can be done without depriv-
ing the equations of their bi-dimensional character.
By putting, in succession, in the terms under the first sign of summation,
t = and i =j, it will be found that these equations contain, severally, the
terms '
[20f — 16; + 2 — 4 (5; — 2) m + 9m^ aoa^^
+ [— ¥ — 8; + 2 — 4 0* — 2) m + 9 m«] a^a,. ,
[— 4f + 8; + 2 + 4 (j + 2) m + Om^ aoa_,.
. +[20/ + 16; + 2 + 4(5j + 2)m + 9m^a^a^,
which are the terms of principal moment in determining a_y and a^ . Let us
then multiply the first equation by
— 4f + 8; + 2 + 4 ( j + 2) m + 9 m%
and the second by
— 20f-\-16j — 2 + 4(5; — 2) m — 9mS
and, adding the products, divide the whole by
48/ [2 (4;-^— 1) _ 4m + m^ .
Then, adopting the notation
r. ri 1 4 0--l)i + 4;-^ + 4;-2-4( i -j + l)m + m»
.'-^'■' j 2(^"^— 1)— 4m + m^
3m^ ^''—8j _ 2 — 4 (j + 2) m — 9m^
W
16^-2 2(4;'^— 1)— 4m + m^
Hill, Researches in the Lunar Theory. 135
, .. _ _3m^ 2()f — 16; + 2 — 4 (5; — 2) m + 9m^
^^^ ~ 1^-2 2(^'^— 1)— 4in + m^
the system of equations, which determines the coefficients a^ , is represented
by the unique formula
2i. [[j, ^i^i-i + Dl a^a-^+y-i + (i) a,a_,_y_i] = 0,
where j must receive negative as well as positive values. It will be perceived
that
hence the last equation is in a form suitable for determining the value of a,-.
The quantities [^',/], [j] and {j) admit of being expressed in a simpler
manner; thus
Fi tl = — i 4- ^^' (•?' ~ ^ i — 1 — m
'■'''-' j^ j 2 (4;-^ — 1) — 4m + m* '
whence
in addition
r • n _L r • n ^* j ^* ( j — i)
L^»J + L— ;.-»J- — j+2(4?-^-l)-4m + m*'
U»*J L ;. *J 2(4f — 1) — 4m + m»'
rfl^ ^^ m' ^ 19f-2y-o-0 + ll)m ,
'■•'-' l(>f 4f 2 (4;^ — 1) — 4m + m'-*
m*
/,-. 27 . 3 13/ + 4?-5 + (5i-ll)m
^''^~ 16/ ^4f 2(4f— 1) — 4m+m^
rfl + (-i)--l 3j+l+2m ,
UJ -t- V 7; — 2j 2 (4f — 1) _4m + m* '
r,1 _ (_,•) _ 27 ^._ 1 16/-3j-5-(%+n)m ,
in V .^^ — 8f 2f 2 (4f — 1) — 4m + m* "
In making a first approximation to the values of the coefficients, one of
the terms of the equation may be omitted ; for, when j is positive, the term
2,. (j) ai&_t_j_i is a quantity four orders higher than that of the terms of the
lowest order contained in the equation ; and, when j is negative, the same
thing is true of 2i. [j]a,a_,+^_,. Hence, with this limitation, the equation
may be written in the two forms
2«- [D', i] a.a,_y + [j] aja_,+y_,] = 0,
2*. [[— j, «] a^+i + i—J) a«a_, +_,_,] = .
where j takes only positive values.
136 Hill, Besearches in the Lunar Tlieory.
From these two equations, by omitting all terms but those of the lowest
order, we derive the following series of equations, determining the coefficients
to the first degree of .approximation,
aoa_i = ( — 1) aoBo ,
a^a2 = [2] [aoai + aia.J + [2, 1] a^a., ,
a„a_2 = (—2) [aoai + a^ao] + [—2,— 1] a^a., ,
aoa3 = [3] [aoao + ^lai + h^^^ + [3, 1] aia_2 + [3, 2] aga.^ ,
a^a_3 = (— 3) [aoa., + aia^ + asao] + [— 3, — 1] a_ ^aa f [— 3, — 2] a_2ai ,
aoa^ = [4] [aoa^ + aia2 + a^ai +a aflo] + [4, 1] aja.g + [4, 2] a2a_2 + [4, 3] aga., ,
aoa_ 4 = ( — 4) [aoa3 + aia2 + ^jUi + agao] + [ — 4, — 1] a_ ,a3 + [— 4, — 2] a_2%
+ [—-4, — 3]a_3ai ,
The law of these equations is quite apparent, and they can easily be extended
as far as desired. The first two give the values of ai and a_i, the following
two the values of a^ and a_2 by means of the values of aj and a_i already
obtained, and so on, every two equations of the series giving the values of
two coefficients by means of the values of all those which precede in the order
of enumeration. A glance at the composition of these equations must con-
vince us that all attempts to write explicitly, even this approximate value
of a,, would be unsuccessful on account of the excessive multiplicity of the
terms. However, they may be regarded, in some sense, as giving the law of
this approximate solution, since they exhibit clearly the mode in which each
coefficient depends on all those which precede it. As to the degree of approxi-
mation afforded by these equations, when the values are expanded in series
of ascending powers of m, the first four terms are obtained correctly in the
case of each coefficient. Thus a, and a_i are affected with errors of the 6th
order, a2 and a_2 with errors of the 8th order, ag and a^a with errors of the
10th order, and so on.
The values of these quantities can be determined either in the literal
form, where the parameter m is left indeterminate, as has been done by Plana
and Delaunay, or as numbers, which mode has been followed by all the
earlier lunar-theorists and Hansen. In the latter case, one will begin by
computing the numerical values of the quantities [j, i], \j] and (;), corre-
sponding to the assumed value of m. for all necessary values of the integers
i and j.
Hill, Researches in the Lunar Theory. 137
The great advantage of our equations consists in this, that we are able
to extend the approximation as far as we wish, simply by writing explicitly
the terms, our symbols giving the law of the coefficients. How rapid is the
approximation in the terms of these equations will be apparent, when we say,
that, after a certain number of terms are written, in order to carry this four
orders higher, it is necessary to add to each of them only four new terms ;
and, thereafter, every addition of four terms enables us to carry the approxi-
mation four orders farther.
The process which may be followed to obtain the values of the a^ with
any desired degree of accuracy, is this: — the first approximate values will be
got from the preceding group of equations until the a,, become of orders
intended to be neglected ; then one will recommence at the beginning, using
the equations each augmented by the terms necessary to carry the approxi-
mation four orders higher; substituting in the new terms the values obtained
from the first approximation, and, in the old, ascertaining what changes are
produced by employing the more exact values instead of the first approxi-
mations. A second return to the beginning of the work will, in like manner,
push the degree of exactitude four orders higher. In this way any required
degree of approximation may be attained.
Whatever advantage the present process may have over those previously
employed is plainly due to the use of the indeterminate integers i and j,
which, although much used in the planetary theories, no one seems to have
thought of introducing into the lunar theory. This enables us to perform a
large mass of operations once for all.
For the purpose of making evident the preceding assertions, and because
we shall have occasion to use them, we write below the equations determining
the coeflScients Ui correct to quantities of the 13th order inclusive.
aoai zz [1] [a? + 2a_,ai + 2'd_.,B,.^ + (1) [aii + 2a,a_2 + Sa^a.g]
+ [1, — 2] a_2a_3 + [1, — 1] a_ia_2 + [1, 2] a2a, + [1, 3] h^^ ,
aoa_i = [— 1] [ail + 2a„a_2 + 2aia_3] + (— 1) [a? + 2a_,a, + 23.282]
+ [_l,_3]a_3a_2+[— 1,— 2]a_2a_,+ [—1,1] aia,+ [—1,2] a^a,,
0082 = [2] [2a«a, + 2a_,a2 + 2a_2a3] + (2) [2a_ia_2 + 2aoa_3 + 2a,a_ J
+ [2,— 2]a_2a^+ [2,— l]a_,a_3+ [2, 1] a,a_i+ [2, 3] a 38,+ [2,4] a^Oj ,
aoa_2 = [ — 2] [2a_,a_2 + 2aoa_3 + 2aia_4] + ( — 2) [2aoa, + 2a_,a.i + 2a_2a3]
+ [-2,-4] a_,a_2+ [— 2,— 3]a^a_,+ [— 2,— 1] a_,a,+ [—2, 1] a.a,
+ [— 2, 2] 828, ,
36
138 Hill, Researches in the Lunar Theory.
8083 = [3] [aj + 2aoa2 + 2a_,a3] + (3) [alj + 2a_,a_8 + 2a«a_ J
+ [3, — 1] a_,a_, + [3, 1] a,a_, + [3, 2] a,a_, + [3, 4] a«a, ,
aoa_3 = L — 3] [aij + 2a_,a_3 + 2aoa_4] + ( — 3) [a? + 2aoa2 + 2a_,a3]
+ [—3,-4] a_,a_,+ [—3—2] a_2a,+ [—3—1] a_,a,+ [—3, 1] a.a, ,
Eoa^ = [4] [2a,a2 + 2aoa3 + 2a_,a4] + (4) [2a_2a_3 + 2a_,a_4 + 2aoa_5]
+ [4,— l]a_,a_5+[4, 1] a,n_3+[4, 2]a2a._2+[4,3] a3a_,+[4,5] aja,,
a«a_4 = [ — 4] [2a_2a_3 + 2a_ja_4 + 2aoa_5] + ( — 4) [2aia2 + 2aoa3 + 2a_,a4]
+ [—4,-5] a_,a_,+ [—4,-3] a_3ai+ [—4,-2] a.^a^^- [—4,-1] a_,as
+ [— 4, 1] a,ar, ,
aoag = [5] [ai + 2a,a3 + 2aoa4]
+ [5, 1] a,a_4 + [5, 2] a^a_3 + [5, 3] aja.^ + [5, 4] a4a_, ,
aoa_5 = ( — 5) [a^ + 2a,a3 + 2aoa4]
+ [—5,-4] a_,ai+ [—5,-3] a_8a2+ [—5,-2] 8.^+ [—5,— 1] a_,a4 ,
aoas = [6] [28483 + 2a,a4 + 28985]
+ [6, 1] a,a_5 + [6,2] 828.4 + [6, 3] 8,8,3 + [6, 4] 848.2 + [6, 5] a,a_, ,
8089 = (— 6) [28283 + 2a,a4 + 2808,]
+ [—6,-5] 8.581+ [—6,-4] 8.482+ [—6,-3] a_,a,+ [—6,-2] 8.284
+ [— 6, — l]a.,a5.
In the first approximation
a, = [1] 80 ,
8-1= (— l)ao,
82 =[l][2(2) + [2,l](-l)]a,,
8.2= [1] [2(-2) + [-2,-1] (-1)] a«,
or, explicitly in terms of m,
_ 3 6 + 12m + 9m* 2
^ -16 6-4m + m* '"*"'
_ 3 38 + 28m + 9m* 2
*-'-~l6 6 — 4m + m* ™ *" '
and, after some reductions,
a2 =^ 2 + 4m + 3m* r238+40m+9m*-32,^ " ^"^.ImX.
^ 256[6— 4m+m*][30— 4m+m*]L ^ ^ 6— 4m+m*J "
a 2 = ?^ 2 + 4.^ + Zrn^ ^T 28-7m + 24 ^ " "^ Im^.
' 64 [6 — 4m + m^] [30— 4m + m^] L ^ 6— 4m + m*J
Hill, Researches in the Lunar Theory. 139
It is evident that, however far the approximation may be carried, the
only quantities, involved as divisors in the values of the a„ are the trinomials,
whose general expression is
2 (4/_l)_4m + ni%
or, particularizing, the series of divisors is
6_4m + m%
30_4m + m%
70 — 4m + m%
It will be remarked that they differ only in their first terms, which are inde-
pendent of'm. Hence any expression, involving several divisors, can always
be separated into several parts, each involving only one divisor, without any
actual division by a trinomial in m. For instance,
1 JL^ 1 l^ 1
[6_4m + m^] [30 — 4m + m^] ~ 24 6 — 4m + m^ 24 30— 4m + m^'
1^ _ 1 1 1 1
[6 _4m + ni^]^ [30 — 4m + m^] ~ 24 [6 — 4m + m^]' 24^ 6 — 4m + m'
^24^ 30 — 4m + m'*
Moreover when, after this transformation, any numerator contains more or
other powers of m than two consecutive powers, it is clear it may be reduced
so as to contain only these by eliminating the higher powers through sub-
tracting certain multiples of the divisor which appears in the denominator,
or, in other words, the fraction may be treated as if it were improper.
From this we gather that the value of a< can be expressed thus
ao~ '^6 — 4m + m2^[6 — 4m + m^2-t-[(._4jn4.ni2]3^- ' •
"^ 30— 4m + m^ "^ [30— 4m + m^]^ "^ [30— 4m + m^]^ "^ ' * '
■^ 70— 4m + m^ "^ [70 -4m + m^]* "^ [70— 4m + m'']" "*" " " '
I ' • • •••••••••
where Jifo, M^ . . . N^, N^ . . P^ P2 - . . are .entire functions of m each of the
form J m* + 5m* ^ ^ .
The advantage of this method of treatment consists in that nothing, which is
given by the successive approximations, would be lost, as must be the case
140 Hill, Researches in the Lunar Tlieory.
when the values are expanded in series of ascending powers of m. The
preceding expressions, when put into this form, become
ax + a_i _ 2 + m ^
— O — III ,
ao 6 — 4m + m^
at — a_ , _ o r 9 _ 4 — 7m "I ^
a, ~ Ls 6 — 4m + nrd°^ '
a 2 + a_2 _ 3 r243 323 + 109m ^^ 23— 11 m 215 — 53m "I ,
ao "" leLie "^ 6 — 4m + nT^ [6 — 4m + n?p 30 — 4m + md ^ '
a, — a_ 2 _ 3 r243 175 + 563m . g 89 — 32 m ^ 361 — 10m 1 ,
ao ~32L8""^6 — 4m-f m'^ [6 — 4m + n?f "^ 30— 4m + m-J"^*
The evident objection to this form for the coefficients is that it makes the
several terms very large, and of signs such that they nearly neutralize each
other, the sum being very much smaller than any of the component terms.
However it may be possible to remedy this imperfection by admitting three
terms into the numerators, but, in this way, the problem is indeterminate,
infinite variety being possible.
It is remarkable that none of our system of divisors can vanish for any
real value of m, since the quadratic equations, obtained by equating them to
zero, have all imaginary roots. In this they differ from the binomial divisors
met with when the integration is effected in approximations arranged
according to ascending powers of the disturbing force.
It is well known that the infinite series, obtained from the development,
in ascending powers of m, of any fraction whose numerator is an entire
function of m, and its denominator any integral power of a divisor of the
previously mentioned series, is convergent, provided that m lies between the
two square roots of the absolute term of the divisor. Hence any finite
expression in m, involving these divisors, can be developed in such a series,
provided that the numerical value of this parameter is less than sf 6 . The
same, however, cannot be asserted when the expression really forms an infinite
series, as it is in the equation just given for the value of — . Yet, on account
ao
of the simplicity with which these quantities can be expressed in this form,
ai and a_i containing each a single term, with an error of the sixth order
only, this limit is worthy of attention.
If the parameter m, hitherto employed by the lunar theorists, is taken as
the quantity in powers of which to expand the value of a^ , we shall have
Hill, Researches in the Lunar Theory.
Ul
m = . And, substituting this value, the principal divisor 6 — 4m + m
1 — m
becomes 6 — 16;/i + llm^ . Thus the limits, between which m must be con-
tained, in order that convergent series may be obtained where this divisor
intervenes, are
^-— . When we consider how little, in the case of our
11
moon, m exceeds m, it will be plain that the series, in terms of m, are likely
to be much more convergent than those in terms of m.
m
If we inquire what function of m, of the form
M
1 +am
, the quantity
[6 — 4m + m^]*
can. be expanded in i)owers of, with the greatest convergency, it is easily
found that a should be ^ • Then putting
m =
m
1+3-"
the divisor 6 — 4m + m^ is changed into
1
and there is introduced the additional divisor 1 + — m . Here the series will
3
be convergent provided m is less than 3. It is true the terms involving the
succeeding divisors 30 — 4m + m^ &c., are not benefited by this change of
parameter, but as they play an inferior role in this matter, I have chosen m
as the parameter for the developments of the coefficients a^ in series of
ascending powers.
To illustrate this matter, we have, in terms of the parameter m, and
with errors of the sixth order,
ai + a_ 1
a
■2+ jin
.l + ii««' i + T'n
m
a,
a_
ao
'5 + -5- m
6
1 + l-^ ««
18
1 + 4- nt
+
27
8
[l + il«]^j
m
2
36
142 Hill, Researches in the Lunar Theory.
Expanding these expressions in powers of m, we get
— - — = — [m + 2 w — -5- at + 86 n» + J ,
^1 *^— 1 11^2 I 6 _^3 1 5 ^4 11
a.
- = T"«^ + l'"' + r2~^-ri «' + ••••
Let these series be compared with those which correspond to them in the
lunar theories of Plana or Delaunay, viz :
19 o . 181 A , 888
m* + - w' + -j^ m* + -2^ m» + ,
^,„. + g,„. + ~„. + 2| ». + ....
The superiority of the former, in convergence and simplicity of numerical
coefficients, is manifest.
Much more might be said relative to possible modes of developing the
coefficients a. in series, but we content ourselves with giving their values
expanded in powers of m, the series being carried to terms of the ninth order
inclusive. The denominators of the numerical fractions are written as pro-
ducts of their prime factors, as, in this form, they can be more readily used,
the principal labor in performing operations on these series being the reduc-
tion of the several fractional coefficients, to be added together, to a common
denominator.
^1 — 3 rr,2 _i_ 1 ,^3 _i_ 7 ^4 _L ^^ rv.'J __ 30749 ^6 _„ 1010621 ^7 _ 18446871 ^g
2io.8».5«
2114667868 9
2ia.8«.6» — • • •
^-j _ _ 19 ,^2 _ 5 m3 ^^ m4 _ 1^ Tr.5 ^881 ^e _j_ 8574168 ^7 _i_ 56218889 _8
ao
^^ ,Ti2 _ ^ m^ — ^^ m^ — ^^ m« _ i!?L tyi« 4- ^^^*^^ m^ -L ^^^1888 9 .
_ m _ - m — ^,-3, m — -- m — ^^^^^ m + 2^:3^6 ^ + 2^8^6^ ^
, 18620168029 ^o
• • •
^2 —26 4 , 803 5 1 6109 6 ■ 897699 7 , 287208647 r 44461407678 9
— 2« •" 27.8.6 "• 25.82.62 "• 2K^Kb^ "" 2i«.82.6* *2i*.8*.65.7 ™ ' • •
^-2 _ Otyi* I 23 5 I 299 -„6 1 56889 ^7 , 79400861 ^g . 8085846888 ^9
---_0m +2,;^m J^-^^—^-m +-^r^,-^,m +2-.32.6^"^ + 2^:8^:6^7°^ ' ' •
ag _ 888 6 I 27943 7 , 12276627 c 1 27409868679 9
a^ "" 212.8 "• 211.6.7 • 210.32.62.72 ^ "• 2i2.84.6».7» "^ • " '
Hill, Researches in the Lumar Theory. 143
■^ — 2«.3 "* ^ 2^3.6 ^ ^ 2».3=».5a.7 "^ ^ 2^3^68.72 "* ' ' *
a4 _ 8687 8 I 111809667 9
^ — "2 iT ^ -r 217.82.6.72 '" • • •
a-4_ 28 •*,8 , 1576663 9
■;;" — 2iV8 "* "^ 2i\8^ Ul . . .
These values being substituted in the equations
T COS v = 2i . a, cos 2it ,
r sin t; = 2i . a. sin 2jt ,
and the parameter changed to m, we get
r cos t? zz
106411 ^6 I 427889 ^7
497664
, 26289037 ^s 782981 ^9 "I ^^^ q
+ 1-4929920 "^ - 873-24800 »^ ' ' * J ^^« ^T
' L266 "^ •" 960 "^ ' 28800 "^ 216000"^
164645868 _8 11821876689 ^« 1 ^
662960000 "^ 19863600000*" * * J
_L r^^^ w6 a. ^Q^^^ tn7 J- 879549 ^g , 181908179 q "l /»
"I" [4096*" "^ 107620*" "^ 1003520 *" "•" 1680644000 "^ ' ' J ^"^ ^'^
+ ^'^«608"^ +9031680^ ...JC0S8t+...| ,
ini;_ao||^-gm ^-^m -h72»» 86^^ 331776 ^^ 276480 ^
269028019 g 1 61872119 9 1 ^\^ O-
74649600 ^ '98812000 ^ .... J Sin ^
4- r^ w^ -I- i?l m^ -I- A^-?i in^ —— m^
» L266 ^ "• 480 ^ "■ 28800 "* 482000 ^
850028 7 8 43885612869 9 1 e,n 4^
11620000 ^ 68660800000 ^ • • • • J ^"* ^
-I- r_!?L tn6 -U 21481 _7 I iil^47^ .„» -I- J598814169 « 1 qin ftr
"^ L12288*" ""mWO*" "" 16052800*" "^4741632000'" • • -J »A" W^
_L r ^875 « I 82608451 « 1 -^ o , |
+ Li966-08^ + 144606880 *» • • • • J S^ 8t + . . . | .
Our final differential equations are capable of furnishing only the ratios
of the coefficients a^, hence we must have recourse to one of the original
r sin
144 Hill, Researches in the Lmtar Theory.
equations if we wish to determine ao as a function of n and \l. By substi-
tuting the values
in the diflferential equation
we obtain
(i^s = ^' •{[<"'■ + 1 + ""'' + 1 "■']"' + 1 ■"'"-- 1 ^" " •
Considering only the term of this, for which i = 0, and supposing that the
coefficient of ^ in the expansion of — ^ is denoted by /, we shall have
For brevity call the right member of this iT; then, since
{■n — nf n^
we shall have
-=m'^n^i-
a •
The value of E is readily obtained from the value of -^ given above, and J
must be found by substituting the values
anW
in —V- ; and taking the coefficient of t. We get
\XiS)\
T _ i , r^^i + ^-HT 3 , 45 rai + a.in^ 15 a,a_, 15 a. + a_2"|
'^-^ + L— a7-J Lt + 64L -T,-"J "^"8 ~V~ 2 ""V"-!
, a^+a_2 r^ a^ + a^g g aia_ H g a^_-f a_ i atag + a_ia_.j
ao L4 ao aj J ao . aji
+ 3 ^^ + 45 ^^ + 3 ^^ ,
a^ fl'* Q*^
<*D <*0
where the terms neglected are, at lowest, of the tenth order with respect to m.
And, explicitly in terms of this parameter,
r _ 1 , 22 4 _ 31 ^.5 _ 68 a _ 2707 7 4201213 » , 14874^89 9
Hill, Researches in the Lunar Theory. 145
By means of which there is obtained
8761 7 4967441 ^g
6912 7962624
or, in terms of the parameter nt,
8 , 11829273 9 -|
^ ^ 39818120 • • • -J '
». = [f]'[l-T»'+^
"•^ 768 *" 6184 "^ 124416 ^^
, j48n_ 7 , %20296 8 I 2?.9^40661_ 9 "I
■" 10368 ^ "■ 71663616 "^ "" 1074954240 ^ • • • J •
The quantity -^ is usually designated a by the lunar-theorists; and,
to make this appear as a factor of the expressions for r cos i; and r sin v, it
would be necessary to multiply all the coefficients by the second factor of the
preceding expression for ao. It seems simpler however to retain a© as the
factor of linear magnitude ; for the astronomers have preferred to derive the
constant of lunar parallax from direct observation of the moon, or, in other
words, they have preferred to consider fi as a seventh element of the orbit ;
with this view of the matter, there is no incongruity in making ao everywhere
replace (i.
The expression for ao can be obtained in several other ways, which lead
to more symmetrical formulae, and which also serve for verification of all the
preceding developments. If, in the preceding equation giving the value of
-r---r in terms of ^, we attribute to r the value 0, or, which is equivalent,
{us)i
make ^ = 1, we shall have t^ = 5 = 2, . a^ , and, consequently
X
= 2,. [(2i + 1 + m)2 + 2m^ a, .
[2i . aj'
And thus, mindful of the value of x given above, we get
_ [jn *
^ = [a
(1 + m)
2
S., [(2,: + 1 + m). + 2m'] ^ . [j, . ?^*_ '
i
Again the differential equation
s+^r:+>=«
87
146 Hill, Researches in the Lunar Theory.
gives
~^y = 2,-. [(2/ + 1 + my — m^ a, sin (2i + 1) r,
7t
and, attributing to r the special value — ,
[2,.(— l)'a,]
Whence
, = 2,. (— 1)' (2/ + I) {%. + 1 + ra) a, .
-m*
(1 + m)^
2, . (- 1)' (2/ + 1) (2* + 1 + m) ^ . [2,. (- !)■• -^T.
ao L aoJ
i
When } zz in the first equation of condition . for determining the coeffi-
cients a,, we get a formula expressing C'in terms of these quantities, viz.,
C = V,. [(2i + 1 + 2m)'^ + i m^] a? + -?- m'^ 2,. a,a.,_, ,
or neglecting terms of the eighth and higher orders,
C=a5ri + 4m + |m^+(9 + 12m+|m^)^J + (l-4m+lm^)^4^ + 9m^^-^l
L ^ -^ aj 2 aj HoJ
1 1 »*
But the C of Chap. I is obtained by multiplyinff this C by - r* = ^ — .
f J 1 J & J 2 2. (1 + mf
Hence, substituting for ao its value, we have
r»_ 1 /„ ,a fl 4- 5m ^ m^ m» ^''® m< "m' ^879 6 1821 t"!
C-^(H>i)i 1^1 + Jm— g m — m —-^^m ___m — j^gjin — .^m J ,
as there stated.
We propose now to reduce the preceding formulae to numerical results.
For this purpose we assume
n = 17325594".06085,
«' = 1295977".41516,
which give
m = -— = 0.08084 89338 08312,
n— n' '
m^ = 0.00653 65500 97941,
m=' = 0.00052 84731 06203,
m* = 0.00004 27264 87183,
m = 0.08308 81293 65.
Hill, Researches in the Lunar Theory. 147
The numerical value of m being substituted in the series, we obtain
a^ = 0.99909 31419 62 [41 * ,
r cos y = a, [ 1 — 0.00718 00394 55 cos ^
+ 0.00000 60424 59 cos 4r
+ 0.00000 00325 76 cos 6r
+ 0.00000 00001 80 cos 8t] ,
r sinu = a^ [ 0.01021 14543 96 sin 2t
+ 0.00000 57148 79 sin 4t
+ 0.00000 00274 99 sin 6r
+ 0.00000 00001 57 sin 8r] .
(To be Continued.)
BIPUNCTUAL COORDINATES.
By F. Franklin, Fellow of the Johns Hopkins University.
The expressions for the bilinear coordinates of a point in terras of its
trilinear coordinates contain a coramon denominator which is a linear func-
tion of the latter. Whenever, therefore, the trilinear coordinates of a point
are such as to make this function equal to zero, its bilinear coordinates are
infinite ; nor are they infinite under any other supposition : and hence the
equation formed by putting this common denominator equal to zero is called
the equation of the infinitely distant straight line.
When we examine the corresponding expressions for the bilinear coordi-
nates of a bm, we do not arrive at any corresponding geometrical idea. These
expressions, too, have a common denominator of the first degree; but the
equation obtained by putting this denominator equal to zero represents sim-
ply the origin of coordinates, a point of no geometrical importance.
I propose to construct a system of coordinates * in which the infinitely
distant point shall hold a position similar to that held by the infinitely dis-
tant straight line in the bilinear system. In the bilinear system we begin by
referring a point to two fixed lines by means of coordinates ; we find that the
equation of a straight line is of the first degree in the coordinates of its points;
and we then define the coordinates of a straight line in such a way that they
will be represented by the coefficients of the point-coordinates in its equation
when put into a certain form. In the system proposed, a line will be referred
to two fixed points by means of coordinates ; we shall find that the equation
of a point is of the first degree in the coordinates of its lines ; and we shall
define the coordinates of a point in such a way that they will be represented
by the coefficients of the line-coordinates in its equation when put into a cer-
tain form, precisely like the corresponding forra of the equation of a line in
the bilinear system.
* When I had written the greater part of this paper, I accidentally discovered, through one of the notes
appended to Salmon's " Conic Sections," that the system of tangential coordinates here presented had been used
before ; where, or to what extent, I do not know.
143
Plate II
:Pi9-^
Fig. 6
Fig 9
Fig. 9
Fig. 4>
Fig. r
Fig. 3
Fig. 5
Fig. 8
Fig. 10
Franklin, Bipunclual Coordinates.
Franklin, Bipunctual Coordinates. 149
Let the points S^ T, (Fig. 1) to which we are to refer all lines, be called
the initials^ and the straight line joining them the base.
The coordinates 5, t^ of a line are its distances from the fixed points /S, T,
measured in a fixed direction — the same for both initials — which will be
called the direction of reference ; the lines passing through S and T in the
direction of reference may be called the lines of reference.
The equation of any point on the base is obviously — zz — - = c. Take
any point P not on the base. Draw PO parallel to the lines of reference, and
let OPzz d. Then for any line of P, we have ^^~-z=.c\ this, then, is the
equation of P. It is an equation of the first degree ; hence the equation of
every point is of the first degree.
Conversely, every equation of the first degree is the equation of a point
For the equation above obtained can be written s-=z ct -\- (1 — c) d\ and any
B C
equation of the first degree As-\- Bt-\- C-=zO can be written s-=z --t j ;
so that, to obtain the point represented hy As -{- Bt -{- C zz 0, we have only to
take a point on the base, such that —- zz j , and from to lay ofi^, in
C^
A r
the direction of reference, a distance ^ or
1-i.l ^ + ^
■^ A
There must, of course, be an understanding as to signs. When 5 or if is
measured upward from the initial, it will be regarded as positive; when
downward, negative. When the distance SO or TO is measured from the
initial in the direction TS^ it will be regarded as positive ; when in the direc-
SO
tion ST^ negative. Thus for all points between S and T the ratio — - will be
negative ; for all points of the base outside of ST it will be positive. The
above equations of points are in accord with this convention as to signs.
I will venture to make a slight innovation in mathematical language. I
have spoken above of the lines of or on a point, meaning the lines passing
through a point; for it seems to me that a closer analogy in our language
respecting the point and the straight line would tend to facilitate both the
comprehension and the remembrance of their geometrical analogies. For
this reason, I propose to call the straight line joining two points their jtmc^^ ;
38
150
Franklin, Bipunctual Coordinates.
for the word intersection is used to designate the common point of two straight
lines, and we ought to have a corresponding word, junction^ to designate the
common line of two points. And as the word intersection is applied to the
common points of curves in general, so the vf or di junction should be applied to
the common lines of curves in general. Thus such expressions as " the
tangents drawn from a point to a curve," " the common tangents of two
curves," &c., would be replaced by the expressions " the junctions of a point
with a curve," ** the junctions of two curv^es," &c.
For the intersection of two lines 5i, ifj, and 53 » Ui we have the equations
As-\- Bt+ C= 0, As, + Bt, + C=Q, As^+ Bt^+ C= 0, whence, elimina-
ting Ay B and C, we have for the equation of the point,
5,^,1
5i , ^i , 1 =0,
^2 , ^2 , 1
If we take as the two lines the junctions of the point with the initials,
we have (Fig. 1) 5, = SM zz a, ^^ zz ; ^2 = 0, ^ zz TN = b ; and the equation
of the point becomes
5,^,1
= ab — bs
or — + ^ — 1 = 0.
a b
a , , 1
0, 6, 1
If we put
1
a
at = 0,
— =z J, this equation becomes
ps -\- qt -{- 1 :=z 0. Let us call p and q the coordinates of the point P ; then
we may regard our equation as expressing either the condition that a line s, t^
should pass through a fixed point p^ q, or the condition that a point p, j,
should lie on a fixed line 5, t. Thus the form of the combined equation of point
and line is precisely the same as in the bilinear system ; and we can save time
and space by adopting at once such of the formulae obtained for the bilinear
system as depend simply upon the form of this equation. Thus we have:
For any line on the intersection of
two lines s^, t, ; ^2, 4 ;
■" 1 + x ' "" i + x *
Si t ti
Hence X =z
02- ^~~ S v2
For any point on the junction of
two points*^,, ji; ^2, ?«;
1 + X
Hence X=^-^^-? = ^
p,—p
1 + X
?2
*It is hardly necessary to mention that the equation of the junction of two points Pn qii p^i q% \ is
Franklin, Bipunctual Coordinates. 151
In the case of the line we see at once (Fig. 2) that X is proportional to the
distance-ratio of the movable line from the two fixed lines ; i. e., the ratio of
the perpendiculars dropped upon the two fixed lines from any point of the
movable line.
In the case of the point, the meaning of /I is not quite so obvious. Rep-
resenting by a, ^, the distances SA, SB, (Fig. 3) cut off on the S line of
reference by the junctions of 1 and 2 respectively with the initial T, and by
p the corresponding distance (SC for any third point 3 of the line 12, we have
Pi = ,^2 = — — ,j) = ; so that
a P P ■
_ P — Pi " P P — "' ^ /^ -^^ ■
a =
P2 — p 1 1^ (3 — pa a' CB
7 1
AC . (13)
Now, wherever the point 3 be taken, -^3 is equal to \-—^ multiplied by a
CJi \y^)
AC '
constant ; so that X^ which is equal to — — multiplied by a constant, is propor-
CB
tional to the distance-ratio of the movable point from the two fixed points.
We have thus seen that the parameter 'k has the same geometrical
significance as the corresponding parameter in the bilinear system ; it is
needless, therefore, to state the theorems respecting the equations of straight
lines passing through a point and of points lying on a straight line, and
respecting their anharmonic ratio ; the theorems and formula) for the bipunc-
tual system will be the same as those for the bilinear system, and we can
make use of those obtained for the latter whenever we have occasion for them.
I shall consider only two metrical problems relating to the point and
straight line ; and these because they will be necessary in the transformation
of coordinates ; and here I shall introduce two words which may be con-
venient. The distance from a point to a straight line, measured in the direc-
tion of reference, will be called the departure of the point from the line or of
the line from the point ; the distance measured on a line parallel to the base
will be called the remove of the point from the line or of the line from the
point. Both these distances will be regarded as positive when the line is on
the positive side of the point
1. Required the departure of a line whose coordinates are given from a
point whose equation is given.
152 Franklin, Bipunctual Coordinates.
Let s', f , be the coordinates of the line, and as -\- bt -\- c =.0 the equation
of the point ; represent the required departure by h. The coordinates of the
line passing through the given point and parallel to the given line are si — S,
t — 5, so that we have
a U — h) +b (t —h) + c = Q^, whence S = ^l±iytf
a-\-o
2. Required the departure of a point whose coordinates are given from a
line whose equation is given.
Let^', 2", be the coordinates of the point, and «p + ftj + c = the equa-
tion of the line; then the coordinates of the line are — and — , and the equation
c c
of the point is ps + q't^ 1 =:0: we have, therefore, from the previous case,
P — + ? — + 1
^ _ c c _(ip + bq^ + c
Transformation of Coordinates.
I. As long as the direction of reference remains fixed, the only change
that can be made in a system of bipunctual coordinates is an alteration in the
position of the initials, and any such alteration can be effected by first moving
them along the base and then moving them in the direction of reference, by
which last process the base itself is moved.
First, let the initials move along the base. Represent by L the distance
ST (Fig. 4) between the old initials, by I the distance S'T between the new
initials, by m and n the removes of the new reference-line at S' from S and T
respectively, and by m' and n' the corresponding removes of the new reference-
line at T\ These quantities are connected by the equations n — 7/i=/i' — vi'=L,
m — mf = n — n! = L We have
Secondly, let the initials move in the direction of reference. Repre-
senting by a and b the departures of the new base from the old initials, we
have obviously s = s^'}-a,t=:f-{-b.
II. When the direction of reference is altered, we have, (Fig. 5)
^ t' sin 3 sin (3' + co — co') . ,. sin (g> — oO
— = — = -T— r , = ^^ r^— ^7 ^ = COS (g) G)') + f^ .
s t sm 3' sm ^' ^ "^ tan 3'
Franklin, Bipunctual Coordinates. 153
XT sl — f sin (cj' — 3') , , sin co' ,
-Now — - — = ^^; — ^ = — cos cj' + , whence
L sin 3' tan 3'
1 S( f '\- L cos 6)'
so that we have
tan ^' L sin cj'
sf t L [sin (J cos (o — cd') + cos cj' sin (cj — cj')] -\- {^ — t') sin (o — cj')
s t L sin cj'
Z sin 6j + (^ — f) sin (cj — co')
Tj sin 6)'
X sin 6)' J. ^, L sin cj'
' i/ sin CJ + (^ — t') sin (cj — cj') ' L sin cj + (5' — t') sin (co — cj') *
If the original system is rectangular and /> zz 1 , these equations become
cosec cj' + (^ — ^0 cot (y ' cosec d + (^ — t*) cot cj' *
The formulae for point-coordinates can be immediately obtained from
those for line-coordinates by observing the eflfect of the transformation of the
latter upon the equation jp8 + |/^ + 1 = 0.
Tripunctual Coordinates.
Just as the non-homogeneous equations of the bilinear system are
replaced by homogeneous equations when we employ three lines of reference
instead of two, the equations of the bipunctual system are replaced by
homogeneous equations when we employ three points of reference.
We may define tripunctual coordinates as follows :
The coordinates of a line are three numbers which are to each other as
the departures of the line from the three vertices of the triangle of reference,
multiplied each by an arbitrary constant.
The coordinates of a point are three numbers which are to each other as
the departures of the point from the three sides of the triangle of reference,
multiplied each by an arbitrary constant.
Let us designate by 5i, ^2? ^3; i>n Ih^ Ih^ the coordinates of line and point,
respectively ; by mi, m,, m^^ the departures of a line from the vertices 1, 2, 3,
of the triangle of reference; and by /f,, Hi^ 71.3, those of a point from the sides
1, 2, 3 ; then we have
These tripunctual coordinates are proportional to trilinear coordinates;
so that from any equation in trilinear coordinates we can obtain an equation
39
154
Franklin, Jiijmncfiml Coordinates.
in tripunctnal coordinates by replacing Wj, ?^2» ^3» A> ^^\i •^a? by ,s, , .s.^, s^^p^^piilh-
For, in the first place, representing by J/j, J/j, J/3, the perpendicular distances
of a straight line from the vertices of the triangle of reference, we have
m^ : /Ho : nh :: M^ : J/, : J/3 ; and therefore if aUi =. XiMi {i zz 1, 2, 3) , we have
only to take (p^ •=. ^X, in order to have /?, : ^2 : 53 :: w, : t/^ • ^^ • And secondly,
representing by N^^ N^^^ iVV the perpendicular distances of a point from the
sides of the triangle of reference, we have X^ = aWi , N^ zz iSn^ , ^3 z= yn^ ,
where a, /£?, and y are constants (viz., the sines of the angles made by the
sides of the triangle with a line drawn in the direction of reference) ; so that,
if ^Xi zz XiNi^ we have ^x^ ■=. ax^Ui , ^x^ => fix^n.z , p^3 zz yx:in^, and we have
only to take \j/, = ^ax, , -^^ m k^jx^ , 4^^ = ^7x3 , in order to have j^i : ^2 • Ih • •
Let us now obtain the equations connecting tripunctual with bipunctual
coordinates.
Let the equations of the vertices of
the triangle of reference be
a^s -\- b^t-}- Cx=0^ where the determi-
028 + bit -f ^2 = y nant r, of the coeffi-
a^H + b^t + ^3 = ) cients, is not zero.
Let the equations of the sides of the
triangle of reference be
Aip+Biq-{'Ci=0 1 where the determi-
J. 2i>+ ^2?+ ^^2=0 >- nant^,ofthecoeffi-
^3^;+i?3g'+C3^0 J cients, is not zero.
The coefficients A^ . . . . C^ are equal (or at least proportional) to the corres-
ponding minors in the determinant of the coefficients a^ . . . . C3 . We have
(page 151)
for anv line .s, t
aiS + ^1^ + ^1
«i + *i
( L^ + bjt + C2
a.2 + &2
«:^ + ^3^ + Cs
nit =
in.,
for any pointy, y,
_ A,p + B,q + C\
n.
C. {p + q)
ni-
(h + *3
Taking ^^=a^-\-b^, (p^.^a^+b^, ^^=a^+b-s,
we have ^1:52: 53 :: a^s + 61^+ c,
: 02^ + M+ ^2 : ^^3^ + M+ ^35
or, and^solvingfor^and^,
^5i = tf,5 + />,/^+(?, y4i^*i + yl.52+^3'^3
—■■ —
IUS2 = «.2S + ^2^ + 6-2 C/,», + Cj^a + ( ^3
6,», + 62S2 + t^
Taking '4'i = Ci, 1^2=^2, ^'3=^3, we have
or, andiSolvingfor^and^r,
vp.i^Aip+ B.jg-\- Ci c,^, + C2i>2+ C3i>3
Cipi+CiP^+CiPi
Franklin, Bipuncfual Coordmates. 155
These equ^-tions are precisely the same as those connecting Cartesian
with trilinear coordinates ; we can, therefore, adopt at once the algebraical
consequences of the equations. Thus we have ^v {p^s^ + ^2*^2 + i^A) = ^ (i^^
+ jif + 1)? and the combined equation of point and line is ^,5, +i^2^2+i^.A = 0.
The bipunctual system may be regarded as a special case of the tripunc-
tual system. Take, first, the coordinates of a line (Fig. 6); designate by s
and t its departures from the vertices 1 and 2, by r its departure from the
vertex 3, and by p the departure of 3 from the line 12. We have ^s^ = 4)1^,
1 T
fiS2 = 4>2^? /w^Sg = <^:jT. Take <^, = 1, ^2 = Ij ^3 = — ; then as^ = s, ^82 = ^, fis^ = — .
P \ P
Now let 3 move in the direction of reference to an infinite distance ; the limit
of the ratio — is unity, and we have s^ : ,%: s^:: s: f: 1.
P ' '
Secondly, take the coordinates of a point. The tripunctual coordinates
of the point P (Fig. 7) are given by vp^ = a//j7i„ v2)2 = ^2^2, vjh = '4''.i^.i^ where 7i,
= -PG,ri2 = — PH,n, = -PK. Take ^|., = -^^-- , ^|., = i^, , .^3=-l, so
that vpx = -—^^ vpo =: — ^, vps=: — /?:{. The coordinates of P in the bipunc-
tual system having A and B for its initials (the direction of reference remain-
ing unchanged) are p = -^— , q= — -— . We have PG : OB : : LGr : LB
and OB: PK::AB: AK, whence Pa = PK 4-~ . ^~ , PH: NA : : Mil: MA
A A LB
and jN'A : PK :: AB : BK, whence PH =1 PK ^^^ . — -r ; therefore vpi =
BR MA
PK AB LG PK AB MH .... ^ , , ^
-LB 'AK'AC^'^'^ = -MA ' BK UC ^ ''^^= '''' ^ow let (7 move
in the direction of reference to an infinite distance ; at the limit, the lines
AC, BC, PG, are parallel, LB is OB, MA is NA, and we have -—^ = - j- ,
LB AB
P^-^ .0 that vp, = -^^,^p, = — M, vp, = PK. But at the limit
MA AB AC JiV
, , LG BK PK .MH AK PK , ^^ ,,, „ PK
we have also _=^^ = ^-^.and-^-^ = ^ = ^; so that .^, = -^,
vp2 = — -5-T^ , vp,, = PK ; that is, p^ :p2 :p^ ::p:q:l.
156 Franklin, Bipunctual Coordinates.
If, in the expressions for s and f (p. 154, end), we put the connmon denomi-
nator equal to zero, we obtain the equation of the locus of all lines whose
coordinates are infinite; that is, of all infinitely distant lines and all lines
drawn in the direction of reference ; and since the equation is of the first
degree, we must regard this locus as a point : the infinitely distant point we may
call it for convenience. Regarded analytically, all lines whose coordinates
are infinite, pass through the infinitely distant point (/. ^., the infinitely dis-
tant point proper to the direction of reference) ; and conversely. Moreover,
we must regard all infinitely distant points as lying on one infinitely distant
straight line ; for the condition that a point should be infinitely distant is
obviously p z=. — j, an equation of the first degree.
We have seen that tripunctual coordinates are perfectly interchangeable
with trilinear coordinates ; it is needless, therefore, to say anything about
transformation from one system of tripunctual coordinates to another. The
coefficients of substitution will have the same geometrical meaning as those
for trilinear coordinates The two systems are, in fact, practically identical ;
whatever can be proved for or with the one can be proved for or with the
other; and it is only in their relations to the bipunctual and bilinear systems
that the distinction between the tripunctual and the trilinear systems comes
into play : tripunctual coordinates standing in the same relation to bipunctual
coordinates as trilinear to bilinear.
Defined as anharmonic ratios, the bipunctual coordinates of a line cor-
respond precisely to the bilinear coordinates of a point :
Designate by X the line from which 1 Designate by Si\ie point from which
X is measured. The number which I s is measured. The number which
represents the ,v of any point is the
anharmonic ratio of four lines passing
through the intersection of X with
the infinitely distant line ; namely,
X^ the infinitely distant line, the line
passing through the point considered,
and a fixed line whose position deter-
mines the unit of length.
represents the s of any line is the
anharmonic ratio of four points lying
on the junction of S with the infi-
nitely distant point ; namely, S^ the
infinitely distant point, the point ly-
ing on the line considered, and a
fixed point whose position determines
the unit of length.
Franklin, Bipunctiml Coordinates, 157
The Conic Sections referred to Bipunctual Coordinates.
Let us see how some of the leading features of curves of the second class,
or conies, present themselves when these curves are investigated by means of
bipunctual line-coordinates. If we take the infinitely distant point as one
vertex of a polar triangle, the opposite side is a diameter of the conic ; let us
designate this diameter as the base-diameter, and the points lying on it as
basics. Any basic being taken as a second vertex of the polar triangle, the
third vertex will be another basic harmonically situated with respect to the
first basic and the intersections of the diameter with the curve. Two such
points will be called conjugate basics ; their analogy with conjugate diameters
may be set forth thus :
Conjugate basics are two points lying
on the base-diameter, so situated that
the junction of each with the infinitely
distant point is the polar of the
other.
All conjugate basics are harmonically
situated with respect to the basics
lying on the junctions of the curve
with the infinitely distant point.
Conjugate diameters are two lines
passing through the centre, so situ-
ated that the intersection of each with
the infinitely distant line is the pole
of the other.
All conjugate diameters are harmoni-
cally situated with respect to the
diameters passing through the inter-
sections of the curve with the infi-
nitely distant line.
Each extremity of the diameter is conjugate to itself; the conjugate of the
centre is infinitely distant. The diameter conjugate to the base-diameter is
parallel to the lines of reference, and joins the points of contact of the two
tangents parallel to the base-diameter.
It will be easy to see what form the equation -of the curve assumes when
it is referred to conjugate basics. (It is always to be understood, unless other-
wise stated, that the direction of reference is that of the conjugate of the
base-diameter.) Let the equation of the curve referred to any triangle be
a^sl + 2ai2SiS2 + ^22^ + ^^^i^^vh + 2«23^.A + ^%A = .
If we put S3 zz 0, we shall obtain an equation which is satisfied by the coordi-
nates of the tangents drawn through 3> and also by all lines passing through
the intersections of these tangents with the line 12 ; for the ratio of Si to s.^ is
constant for all lines passing through a fixed point on the line 12. Now, if
40
158 Franklin, Bipunctaal Coordinates,
our triangle is a polar triangle, these intersections are the points where the
line 12 cuts the curve; they are therefore harmonically situated with respect
to the points 1 and 2, and the equation which represents them (viz., the equa-
tion ^ii'^i + 2«i.2.Si82 + a.22."^.] zz 0, obtained by putting s^ = 0) must be of the form
SI — 7.sl =. ; that is, the term containing s^Si can not appear. It is evident,
in the same way, that the terms containing ^1^3 and 8.28^ can not appear ; so
that the equation of the curve referred to any polar triangle is of the form
aisl + a-yd + a,i4 =■ 0. If, then, in this equation, we replace ^1 by 5, ^2 by f,
.% by 1, we shall have the form of the equation of the curve referred to con-
jugate basics which may therefore be written -- + -— = 1 * Either c^ or d^
(T d
may be negative, /. ^., either c or d imaginary ; they cannot both be negative
unless the curve itself is imaginary.
We see that for any value of s there are two equal and opposite values
of t] L ^., the two tangents drawn from any point of either line of reference
cut the other in two points equidistant from the base. This geometrical
property follows immediately from the definition of Conjugate basics; and we
could have inferred the form of the equation from this property, instead of the
converse.
Before going any further, let us see how this simple form of the equation
is derived from the most general bipunctual equation of the curve a^s^ -{- 2ai2^t
+ «22^^ + '^^vs^ + 2^23^ + ^33 = 0. As tripunctual coordinates are interchange-
able with trilinear coordinates, and are replaced by bipunctual coordinates in
the same manner as trilinear coordinates are replaced by bilinear coordinates,
we can here use at once certain algebraic results obtained for the latter (see
Clebsch, p. 82, et seqq,) The coordinates of the base-diameter are
^ - ^13 ^1^ ^^1 -^23 ^1^12 ^1^32 .
-0.33 ^n^h2 — ^2 -^33 ^n^h2 — ^
2
12
and I shall suppose, for the present, that .4 33 is not zero. To transfer the
initials to the base-diameter (the initials moving in the direction of reference)
we put 5 zn s' + (T, ^ — ^ + T, and our equation becomes a^sf^ + ^^^ + 2^125'^
-f — zz 0. (It is supposed throughout that -4, the determinant of a,i . . . . a^
-4 33
is not zero.)
* In the above reasoning, as in general where the case has admitted it, I have followed the method used in
Clebsch 's ** Vorlesungen uber Oeometrie."
Franklin, Bijyunctual Coordinates, 159
If we represent by m and n the removes of a straight line from the initials
S and T, and by 3 the ratio of the sine of the angle the line makes with the
base to the sine of the angle it makes with the lines of reference, we have
5 =z — m3, tzz. — 71^ \ also 71 — mzz Lj where L is the distance between the
initials. When the line considered is parallel to the base, 3 = 0; when it is
parallel to the lines of reference, ^ = co .
The condition that a line s, t, should pass through the pole of the line
A
sfj t\ with reference to the curve «n^ + ^22^^ + 2«i25^ + -j- = 0? is a^ssf + ^22^'
-4 33
A
+ a,2 {sf + ^t) + -J- = ^ ; when the two lines are parallel to the lines of
Ass
reference, this equation becomes (replacing s by m^, t by ?i3, «' by m'3, and
f by 7i'S', and dividing by 3, and then making 3' infinite)
a^{inm' + a^^^nri' + a^^ {^^^^' + ^^'^) ==0, (1)
or, substituting L + ^>^ for ;i, L + m' for n\
{an + 2«i2 + «22) >>«>^^' + («i2 + ^22) (^ + ^') L + ^22/^^ = .
If we transfer the initials to points whose distances from the former
initials are m, n, and m', n\ respectively, we shall find that (1) is the necessary
and sufficient condition of the disappearance of the term containing st To
Tilt — ~ 7)1*8 Tit -~— Tl!s
effect the transformation, we replace s by and t by j — (where
lz=.m — m' zz n — Ti is the distance between the new initials) ; and the
equation becomes {a^^n''^ + ^a^^^n'Ti' + a^ii^) s^ + {an^n^ + 2a^2'^Ti + ^22^^) t^
— 2 [aiimm' + 022^^' + a^o {mnf + m'n) ] st + — - P zz . (1) expresses the
-4 33
necessary and sufficient condition of the disappearance of the term containing
st', when that condition is fulfilled, our equation becomes (a,iW'^ + 2ai2m'n'
A s^ t^
+ 022^'^) s^ + (autn^ + 2^12^71 + ^22^^) t^ + --P = 0, or -- + — z= /% where
Ass ^ ^
^ — A . - and d'^= — . -
^33 ' aiiTn!^ + ^duPfiTtl + ^22^'^ ' A^ ' «,iW^ + 2^12^^ + ^^"^
In the equation — -^ — tn P , c^ and rf^ are real quantities, but not neces-
s^ s^ I c^d^
sarily both positive. For 5 zz ^, we have _ . + __ zz /% whence 5 zz± l\-^ — ^7; ;
cr d^ (T-^-d^
160 Franklin, Bipunctual Coordinates.
hence the length of the diameter conjugate to the base diameter is
:^ . Putting — u^ for s and — v^ for t m the equation of the curve,
(? + cP'
2^2 ^2^2 '1 ^2
it becomes ^— -| — — = P ; when ^ zz oo , we get -^ + — = ; or, putting
(f ' d' ~ ' ' ° cr^ ' rf
,2.72
for V its value / + ju, -^ + ^ ^^^^ zz 0, whence (i = I ^-rin
The difference between the two values of ^ is the length of the base- diameter,
which is therefore ± 2Z . Representing by 2a, 2ft, the lengths of
the base diameter and its conjugate, respectively, we have, then, a^ zz
-^\-^-,Zy' *■^ = ^M^^' whence 0^ + ^^^ = - -^ , <^^' — :^r ^° *^^*
(? and rf^ are the roots of the equation or^ H — ^ x -~ zz , and their values
cr arl^
are - .^- {I ± V7M-~ 4^^) •
When 5 rz 0, ^ zz ± W ; when if zz 0, 5 = ± fc ; that is, the departure
from aS of a tangent passing through T is Ic^ and the departure from T of a
tangent passing through S is M; these distances may be called the initial
departures of the curve. We have Pc^ + Pdrz=: P, J^c^dj^ = -P. Now
a^ a^
a and 6, being the lengths of the base-diameter and its conjugate, are unal-
tered when the initials are moved on the base; hence, whenever the initials
are conjugate basics on a given diameter, the sum of the squares of the initial
departures is proportional to the square of the distance between the initials,
and the product of the initial departures is proportional to the distance
between the initials. The initial departures cannot both be real unless the
initials are on a diameter which cuts the curve in imaginary points and whose
conjugate cuts the curve in real points ; for it is only when b^ is positive and
J2
d^ negative that the expressions for c^ and rP, namely — — -;^_ (l^ VP-\- 4a^),
• 2a I
are both positive.
Let us see how our curves can be classified. We have
1) When c^ -\- (P = 0, a and b are both infinite.
2) When c^ + d^ is negative, a and b are. real and finite.
3) When r + d^ is positive, a is imaginary and b real if c^ and /P are both
positive ; a real and b imaginary if c^ and d^ have opposite signs.
Franklin, Bipunctual Coordinates. 161
(When (? -\- d^ zz. — 1, a and b are equal in absolute value and both real ;
when c* + ^ = + 1> «^ a^^d b^ are equal in absolute value, but either a or ft is
imaginary.)
It would seem, then, that we can divide our curves into three classes :
1) Those in which the base-diameter and its conjugate are both infinite in
length, (c^ + rf2 = 0).
2) Those in which they are both terminated in real points (^^ + ^ < 0).
3) Those in which one of the two diameters is terminated in real points
and the other in imaginary points (c^ + (P > 0).
But we must see whether these characteristics are preserved when the
direction of reference is changed ; or in other words, whether, if one pair of
conjugate diameters belongs to a certain one of these categories, every pair
of conjugate diameters of the same curve will belong to the same category.
Returning to the expressions which we replaced by c^ and eP (page 159), we
have
c^ + (?-._ j4_ / 1 ^ 1 \ Thequan-
-4 33 \a^xm^ + 2ai2mn + a^^v? a^xni''^ + 2ai2^n'n' + (h&n'^^
tity m the parenthesis is equal to -^ — r- ^ ^^ ^— --^V s^ — -'- ^..
{aiyin^ + 2a^2^n + a^inr) {a^y^m! ^ +2ai2mfn' + ^22^)
If we subtract from the denominator of this fraction the square of aumm'
+ chi^n' + ^12 (^^' + M'n), which is equal to zero, and subtract from the
numerator 2 [oninrn' + (Z22^m' + ai2 (mw' + mfn)'], the fraction reduces to
«ii (^ — ^f^y + 2ai2 {fn — mf) {n — n') + ^22 (^ — ^0^
(^11^2 — ^12) (^^^' — mfny
or, since m — m' z=z n — n' zzl and n — mzuvl — m' m //, to " -.^^±—
so that we have
22
33-
A-'iAij
c^ -^A^ — — -^^-^ («!! + 2ai2 + ^22) .
33-
It will be convenient to* postpone the investigation of the change which this
expression undergoes when the direction of reference is changed ; but I must
anticipate the results of that investigation so far as to say that A becomes aM,
where a is a real trigonometrical function which cannot vanish; and that
«!! + 2^12 + ^ remains entirely unchanged. So we see — since the denomi-
nator is essentially positive — that the sign of the above expression is unal-
41
162 Franklin, Bipunctual Coordinates.
tered by a change in the direction of reference ; and that if it is zero for any
direction of reference, it is zero for all. We have, then, the general classi-
fication :
1) a„ + 2^12 + ^22 is zero ; all diameters have infinite length (parabola).
2) «ii + 2«i2 + ^22 has the same sign as A ; all diameters have finite
length (ellipse).
3) ^n + 2^12 + ^2 has the sign contrary to that of ^ ; of each pair of
conjugate diameters, one has real length and one imaginary length (hyper-
bola).
We have found & -\- d'^ •=. — -127^2 (^^ + ^^2 + ^22)) and we have also
^ For
c^d'^'=i—-, _
^33 («ii^H^ + 2ffi2m7i + a<22V?) {aiimf^ + 2a^2'^M + ^22^'^) Al^UP '
any given direction of reference, all the quantities involved in these values,
except /, are constant ; so that we see again that {c^ + d^) f^ is proportional to
P and that cdP is proportional to /.
Let us investigate the condition of the equality of conjugate diameters.
A
That condition is expressed (see p. 161, 1. 1) by the equation — -T2-T2 (^1 + ^^12
A^L
+ ^22) = — 1- If we represent by ^ the angle made by the original base with
the lines of reference, and call the distance between the original initials
unity, we have, since — — and — — are the coordinates of the base-diameter,
-^33 -^33
Z* = 1 + ("^^ "" "^^^V 2 ^i" 7^^ cos <^ ,- and the equation of condition
33 ' -^-^33
becomes j\\ , ^'] "^ ^^ -^-^ = 1.— Let the equation
Al, + ( Jia — ^23) + 2^33 (^13 — ^23) COS ^
of the curve referred to a rectangular system be
anS^ + 2ai2st + a22t^ + 2a,35 + 2a2zt -f ^33 zi: 0.
Let the direction of reference be altered so that the lines of reference shall
make an angle 0/ with the base ; denote the coefficients of the new equation
by fl'u, &C.J their determinant by .4', and its minors by A\i, &c. The trans-
formation is effected by means of the formulae (p. 153) s = -— ,
^ VF / a + /8(5'— ^)'
Franklin, Bipunctual Coordinates, 163
t = rr-, ; where a = — and 3 = . — , so that cos o' = — and
a + p {af — f) sin o' sm w' a
a* = 1 + /?^ We have
«'ii = ^11 + 2^13/3 + ^33/32 , a'22 = «22 — 2^23/? + %3/3^ , a'33 = agga^ ,
^12 = «12 («13 «k3) /? «33/3^ , a'i3 = O^^ + ^gsa^ , a'23 = «23a ^SSO^i^ )
whence we find
an + 2a,^ + a^' , ^12 — («i3— ^23) /? — ^33.^" , «i3a + «33a/?
«12 («13 ^23) /3 «33i^^ , «22 2flf23/3 + «33/3^ J ^230' (^3^t^
«i3a + + ^33^/3 , a23a +0 — a33a/3 , + a33a^
^'33 = ^33 - 2/3 {Ars — A^) + (3' {An + ^.2 — 2A,,) = A^, - 2^G + /3W,
^'13 — -^'23 = —a(3 {An + A^ — 2^12) + a M13 — ^23) = olGt — a/3//,
(where, for brevity, we put A^^ — ^23 = Gr , An + ^22 — 2^12 = H) and
a'n + 2a'i2 + <^22 = ^11 + 2^12 + ^22- If? i^ow, in the equation
A (a'n + 2a^,, + aVz)
^' =
= aM,
^'33 + (^'13 — ^'23) + 2 J'33 {A\, — A',,) cos 4)
= 1,
we substitute the values above obtained, and substitute — (zzcosa)')forcosd),
a
we obtain, after reduction,
Al + G' — A {an + 2a,, + a,) — 2G {A^ + H) ^
+ iG'+H'—A {an + 2a,, + ^22)] /3^ = 0.
From this equation we can, in general, obtain two values of /?, which give us
the directions of the two equal conjugate diameters.
In order that every diameter should be equal to its conjugate, the above
equation must be satisfied for all values of /3, and we must have
Al+G' — A{an + 2a,, + a^)=0, G{A^ + H)=0,
and G"" + H^ — A {an + 2ai2 + ^22) = 0.
Substituting for G its value, and remembering that ^an = ^22^33 — Al^^ &c.,
the first equation becomes 2 {A,^ — ^23)^ — ^33 {An — 2^12 + ^22 — ^33) = ,
or 2G^'— ^j^3(//— ^33)=0, and the third becomes likewise 2 (x'—^ 33^+^2^ 0,
or 2G^ + H{H— ^33) =0. We have, then,
(1) . . . . 2G' — J33 {H— ^33) = 0; (2) .... G^ (//+ A.^,) =0;
(3) 2G' + H{H— A^) = 0.
Equation (2) shows that we must have either 6^ = or /f + A^=z 0. In the
first case, we must also have H — ^33 = in order that equations (1) and
164 Franklin, Bipunctual Coordinates.
(3) should be satisfied ; in the second case, equations (1) and (3) become
G^ + ^33 1= 0, which cannot be satisfied if the coefficients of the given equation
are real, as we suppose them to be throughout. The necessary and sufficient
condition, therefore, that every diameter should be equal to its conjugate is,
in any rectangular system of coordinates, (the distance between the initials
being taking as unity)
G= and // — ^33 = ; that is, Ai^ = A^^ and -4n — 2^12 + -^22 = -^gg.
The square of half the diameter making the angle cot""*/? with the origi-
nal base is (pp. 160, 162)
II ^^ ^ <^ .
(^ + ^~ ^'33 (a'n + 2a!n + aW "" (^33 — 2/?G^ + ^'H) {a,, + 2a,, + a,,) '
when G = and H = A^<i, this becomes
a^A A
(1 + (3'') ^33 («ii + 2a,2 + «22) ^33 («ii + 2ai2 + ^22) '
an expression independent of ^. So we see that when every diameter is equal
to its conjugate, all diameters are equal ; i. e.^ the curve is a circle. The con-
dition Axz = ^23 shows that the base-diameter is parallel to the original base ;
we have seen that this condition holds in the case of the circle for any rectan-
gular system : therefore, in the circle, every diameter is perpendicular to its
conj ugate.
The condition & -^^ 6?=l-\-\ (see p. 161, 1. 2) would become, in the same
way,
-(l + /?^)^(an + 2a,2 + a22)=^^+G^'-2<y(^33 + ^)/?+(G^ + ^^)/?^
or ^^+ G^' + ^ (an+ 2ai2 + «22)— 2(y (^3 + ^^) /?
and we should find, in like manner, that if this equation is satisfied for all
values of ^ we have J33 (//+ ^33) = 0, 6^ (iT-f ^3:*) = 0, and H{H'\- A^ — 0,
so that the only condition here is ^ + ^33 = 0-
It may be worth while to observe that the results obtained for A' , ^'33
^'13 — ^'235 and a'u + 2a'i2 + a'ga, are entirely independent of the fact that the
original system was rectangular; they apply to all cases: and it is only in the
replacing of a'^ by 1 + /?^ and of cos 4) by — - , that the rectangularity of the
original system comes into play, and simplifies the results.
Franklin, Bipunctual Coordmates, 165
The angle between the base-diameter and its conjugate (Fig. 8) is given by
«5"^ 1 - ^- -whence tan 5= ^ ^ '^" "'
sin (6)' — h) ^ J13 _ ^i? A\3 — ^'m ^'i3— ^'28 + ^'SS COS O'
^33 — 2^G + /3W
; so that we have the equation
tan«5 ^ 4 ""^'
^'{H+ a tan 5) + /? [(ir— ^33) tan 5 — 2(?] — G^ tan 5 + ^33 = 0.
The condition for the reality of the values of /? is
4:{A^—a tan h){H + G tan S) < [{H — A^) tan 5 — ^Gf ,
which reduces to
A^H— G^_
> ^ {H— A^' + 4G '
This condition is satisfied for all values of 8 if A^E — G^ is not positive, i. e.
if ^33 {All — 2^12 + -^22) — (^13 — AisY < , and this can be put into the
form ^11-433 — A\s + A^A^ — A%+2 (^13^23 — ^12-433) .< , or ^ {an + 2ai<^
+ ^22) < 0. Thus we see that in the parabola and hyperbola the conjugate
diameters may make any angle with each other, but that in the ellipse there
J IT /J2
is a minimum angle given by the equation tan*S = 4 ^ ^ -Tg r^r^ • ^^
{M — ^33) + 46r
the case of the circle, H — ^33= and G^zzO, and this equation becomes
tan*5 = 00 , whence we see again that in the circle every diameter is perpen-
dicular to its conjugate.
Solving the quadratic in /? found above, we obtain .
^ _ — {H— ^33) tan 8 + 2G± Vl{H— A^^f + 4W'] tan'^ g -- 4 JA^H— ffl)
Since A^H — G^'=:A {a ii-\- 2ai2 + (122)^ we have, in the case of the parabola,
As^H — (r* = 0, and the values of /? will be found to reduce to
n _ Ass — G ta n 5 , o _ ^^s
The second of these values is independent of S; therefore in the parabola there
is a fixed direction in which one diameter in every pair runs. In order that
this direction should coincide with the direction of reference we must have
.42
166 Franklin, Bipunctual Coordinates.
/(3 zz 0, whence A^ zz 0. Now, when A^zn 0, the coordinates of the base-
diameter are infinite; and since we have seen that the conjugate diameters
may make any angle with each other, this cannot be interpreted to mean that
the base-diameter is parallel to the lines of reference, i e. to its conjugate: it
must, then, be infinitely distant, and we have therefore found that in the
parabola all diameters that are not infinitely distant are parallel to each other.
Let the equation of a conic referred to a certain pair of conjugate basics
s^ t^ .
be — + — iz //^ and its equation, referred to any other pair of conjugate
basics on the same diameter, -- + -— zz IK Putting C^ -{- D^ =: g, we know
r d^
that, wherever the new basics be taken, we have c^ + (f^zz^ and l^(?d^z=,I?C^iy^\
and if we assign any particular value to /, e, or r/, we can obtain the values of
the other two of these quantities from these equations. If we put rf^ zz — 1,
so that the equation of the curve becomes ^ •=. c^ (t^ + Z'*), we have c^ zz ^ + 1,
Pzn U. The condition that the basics for which the equation of the
curve assumes this form should be real points is, that this value of V^ should
be positive ; and it is easy to see when this condition is fulfilled :
In the parabola, CIJ^ is negative and y = ; hence V' is positive.
In the ellipse, C^UF is negative and g is negative ; so that V' is positive
when g is numerically less than 1, and negative when g is numerically
greater than 1 ; that is (p. 160), positive on a diameter that is greater than
its conjugate and negative on one that is smaller than its conjugate.
In the hyperbola g is always positive ; C'^IP' is negative when the diam-
eter cuts the curve, and positive when it does not ; in the former case l^ is
positive, in the latter, negative.
Of course, to every numerical value of I there correspond two pairs of
conjugate basics, symmetrically situated with respect to the centre ; in the
parabola, one of these pairs is infinitely distant.
It appears from the discussion of the angles made by diameters with
their conjugates, that every conic has one pair of rectangular conjugate
diameters, or axes ; and we now see that on one of the axes there can Vie
found, on each side of the centre, a pair of conjugate basics for which tlie
equation of the curve assumes the form ^ = e^ {f^ + V^) ; that is, a pair of
Franklin, Bipunctual Coordinates. 167
conjugate basics, S and T, such that the ratio of the distances from S to the inter-
sections of any tangent with the S and T lines is constant It is plain that ^ is
always positive, and that in the parabola e* = 1, in the ellipse e^ < 1, and in
the hyperbola ^ > 1.
Let us see how, by means of this property, we can construct the tangents
to a conic from a given point
For the parabola, the construction is very simple. Let S and T (Fig. 9)
be the basics in question, and Jf the given point. The line drawn from S to
the point midway between the intersections of the tangent with the S and T
lines is perpendicular to the tangent (since aS' is equidistant from those inter-
sections) ; that point is consequently on the circle described upon SM as a
diameter. But it is also on the line drawn parallel to the S and T lines
midway between them ; it is therefore at the intersection of this line with the
above-mentioned circle. Hence, to construct the tangents from M^ we draw
OP midway between the S and T lines ; and from N^ the middle point of MS,
we describe a circle with NS as radius ; the lines joining M with the inter-
sections of this circle with OF are the required tangents. If NO = NS^ the
tangents coincide and -M is a point of the parabola ; and conversely, if M is
a point of the parabola, the tangents coincide and NO = NS. But -^7,= ~i',
NS MS
therefore, the distance of any point of the parabola from S is equal to its
distance from the T line. When MQ is greater than MS, the two tangents
are imaginary ; when MQ is less than MS, the two tangents are real and
distinct ; when M is on the T line, the two tangents are perpendicular to each
other. We may call S the focus and the Tline the directrix of the parabola.
Let us proceed to the ellipse and hyperbola. We have, for any tangent
MO (Fig. 10), -~ = e. Denote the distance ST by Ef, where E = -^- , take
SP zz /, and draw PQ parallel to the lines of reference; it intersects SO at a
point Q. . It is obvious that SQ m SN, and QN is a tangent to the parabola
whose focus is S and directrix PQ, If, therefore, we can determine the point
U where QN intersects SM, we can construct the tangents from M by con-
structing the tangents from U to the parabola (SP) and joining M with the
intersections of those tangents with the aS' line. Denote the distance SM by
Ez and take SB = z, PQ is parallel to MO; hence, if we draw any line MV^
meeting the S line at V, and a parallel line PW meeting PQ at W, U is the
168 Franklin, Bipunciual Coordinates.
intersection of FPFwith SM. We have, then, the following construction for
the tangents to the ellipse (or hyperbola) from the point Jtf, the curve being
given by 5, T, and P. Through M draw any line meeting the S line at V\ on
SM take a point R such that — — = ■— , and through It draw a line parallel
to ^fV, meeting the P line at W] join VWy and from f/, the intersection of
FTFwith SM, construct the tangents to the parabola (SP) ; the lines joining
M with the points where these tangents cut the S line are the required
tangents.
Let us take MV and BW parallel to ST, and produce MV to meet the T
line in X; also draw i7y parallel to A^Tand meeting the P line in Y. It can
r/Si i/^s
be shown* that -r?^^ = E -—- ; whence it follows at once (from what we
(11 MX
have found for the parabola) that the distance of any point on the ellipse (or
hyperbola) from the focus is equal to e times its distance from the directrix —
S being again designated the focus and the T line the directrix ; also that no
tangent can be drawn when the ratio of these distances is less than e, and two
distinct tangents when it is greater than e.
It is plain that in the ellipse the foci lie between the extremities of the
axis and the directrices outside of them, and that the reverse is the case with
the hyperbola ; it follows immediately from the above, therefore, that in the
ellipse the sum of the distances of any point from the two foci is constant, and
in the hyperbola the difference of the distances of any point from the two foci
is constant.
The following are obvious consequence^ of the property by which the
focus and directrix were brought to our notice: in the parabola, any tangent
makes, at its intersection with the directrix, equal angles with the directrix
and the line joining that intersection with the focus ; in the ellipse and hyper-
bola, any tangent makes, at its intersections with the two directrices, equal
angles with the lines joining those intersections with the corresponding foci.
* PenoUi MX by Ey ; then It W = y. Wo have
J5/— (J5;— l)y' SM EJ-(E—'i)y' Ez - Ef — (E — I) y '
fy ^ . SU Ez „Ez „SM
•^^ ■ therefore - „ = ~ = E ^^ = E ~~ .
E/^(E^l)y' Ur~ y - Ey- MX
Franklin, Bipunctual Coordinates. 169
Let j^^ — zz L^he the equation of a conic referred to any pair of con-
jugate basics on an axis; is there a point on the axis the ratio of whose
distances from the intersections of any tangent with the S and T lines is
ri2
constant ? Writing the equation in the form s^ = — t-o*^ + ^^^ j we see
that if such a point exists, then, denoting its distance from S by aL (whence
its distance from T is (1 + a) Z), the above equation must be identical with
(P
^ + q}U = — [^2 _|_ ^j ^ ^^2 jr2j . gQ ^^^^^ ^g have, for determining a,
^' (1 + a)' + a' = — C' , whence
_ —C^±V— CD' {C^ + U^ + l) _ — C^ j- V _ (PD' {g + 1)
"*"" (P + B' ~ g
It is plain that the condition of the reality of these values is the same
as that of the existence of real foci (see p. 166, 1. 14) ; so that points possessing
the property in question can be found only on the axis on which the foci are
situated. The distance from S of the point midway between the two points
0^ _L ot * C^
corresponding to the two values of a is -^—~ — ? L zz. Z , which, it will be
2 g
readily seen from p. 160, L 4, is the distance of the centre ; so that the two
points are equidistant from the centre. The distance between the points is
(oi — og) L = ^^—^ — ^ =: ^^-^ — ^ = — (smce fl' + 1 = ^) ;
ff 9 9 .
and this is obviously equal, in absolute value, to twice the distance from the
focus to the centre ; that is, equal to the distance between the foci. Hence,
the points sought are the foci themselves, and we have the following theorem
(in which the term ''conjugate ordinates" is to be understood as meaning
ordinates to the transverse axis passing through conjugate basics on that
axis) :
If M and M' be the points in which any tangent to a conic intersects a
fixed pair of conjugate ordinates, and F a focus of the conic, the ratio — — is
constant, and the same for both foci.*
* This theorem admits of several easy geometrical demonstrations.
43
170 Franklin, Bipunctiial Coordinates,
It may be remarked that this ratio, — , is unity for all pairs of conjugate
ordinates in the parabola ; in the ellipse and hyperbola it varies for different
pairs, from to 1 or from 1 to oo .
The following are one or two obvious consequences of this proposition :
1. Since the ordinate conjugate to the minor axis is a straight line per-
pendicular to the transverse axis and infinitely distant, and since the distance
from the focus to tlie intersection of any tangent with this infinitely distant
ordinate is proportional to the secant of the angle made by the tangent with
the transverse axis; it follows from the theorem just established that the
distance from the focus to the intersection of any tangent with the minor axis
is proportional to the secant of the angle made by the tangent with the trans-
verse axis. In the case of the ellipse, this distance is equal to half the
transverse axis when the angle is ; so that the distance from the focus to
the intersection of any tangent with the minor axis is equal to the distance
intercepted by the axes on a line drawn through an extremity of the trans-
verse axis, parallel to the tangent. It can easily be seen that this result is
true of the hyperbola also, within the limits to which its real tangents are
confined.
2. Let M be the point of contact of a tangent, and M' the point at which it
"PlUf TC'f lif
intersects the transverse axis. We have, by the above theorem, -— - zz — — ,
whence we see that the angle F'MM' is the supplement of FMM' ; that is, the
tangent makes equal angles with the lines joining its point of contact with
the foci.
Finally, let us ascertain whether the foci or any other points enjoy a
property like that which we have been considering, with respect to any pair
of parallel lines other than conjugate ordinates. Refer the conic to any
rectangular system with the given parallels as lines of reference. Denote by
a the distance from the base of a point possessing the property in question,
by b its distance from the S line, by L the distance between the parallels, and
by m an indeterminate constant ; then the equation of the conic is
(s — af + W = m i{t—af + {b + i)^ , or
52 _ ^^^f.2 _2as + 2mat + a" + b^ — m [a" + {b + Z)^ z= 0.
Hence we have, with the usual meanings for Aiz &c.,
A A
^13 = — f)iay A2S = — ma, A.^ = — m, -J^ zz -^^ zz a ;
Franklin, Bipunctual Coordinates. 171
and it follows at once (see p. 158, 1. 26) that the parallels are perpendicular to
an axis and that the point or points sought lie on the axis ; and the absence
of the term containing st shows that the parallels cut the axis in conjugate
basics. We are thus brought back to the case already investigated; and
conjugate ordinates are therefore the only pairs of parallels, and the foci the
only points, enjoying the property in question. That it cannot belong to lines
not parallel is evident.
The Conic referred to its Foci. — The distances of the extremities of the axis
from the S initial* are (p. 160) I ^ — — , so that the distance of the
centre from the S initial is -. ; and in like manner the distance of the
r + »
. . . . Id^
centre from the T initial is -. -. . When the S initial is a focjus, these dis-
e^l I
tances are x , x ; so that the distances of the other focus from 8 and
1 — r 1 — r
T are ^ and ^ ' ., I. To transfer (see p. 152) the initials to the foci, there-
1 — e^ 1 — ^
It- \±l Is
1 ^^ n e^)t (l4'e^)s
fore, we leave s unchanged and replace t by —^ zz ^^^ ^ y ' ^ ,
and the equation s" =z e" {t^ + P) becomes 4^$" = (1 — e'f #' + (1 + e'y s^
— 2 (1 — ^) st+4^P ors^+ #2 — 2 J-t^, st + _if-^ = 0. The last term
1 — e^ (1 — ry
in this equation is the square of the distance between the foci ; denoting that
distance by 2 k^ and "* by h, the equation becomes
X ^^ e^
^^t^ — 2hst + 4K' = 0.
The equation of any other conic having the same foci would be s^ + ^^
— 2h!st + 4# zz ; combining this with the preceding equation, we have
5^ zz for the common tangents of the two curves. This condition is satisfied
only by lines passing through the foci ; and the equation of the curve shows
that tangents passing through the foci are imaginary. Hence, all confocal
conies have in common four imaginary tangents passing through the foci.
Denoting by c, tt, the lengths of the perpendiculars dropped from Sj T,
upon the line s, #, and by p the cosine of the angle made by the line with the
172 Franklin, Bipunctual Coordinates.
base, we have szz — , t-=z — , p^zz — ^ ^ . By means of these equa-
tions we can transform any equation in s, ^, in a rectangular system, into an
equation in cr, r ; and the equation of the conic referred to its foci assumes,
when thus transformed, a very simple form. We have, first, a^ + r^ — 2hjar
+ 4Fp^ zr ; but f zz avT " ' ^^^ *'^® equation reduces to 2 (1 — h) at
+ 4^ zz 0, or ar zz - — - zz ^ = ¥ , where h is half the minor axis (see
h — 11 — e
p. 160, 1. 8). Hence, the product of the perpendiculars dropped from the foci
upon any tangent is equal to the square of half the minor axis.
To remove T to the centre, we have simply to replace ^ by 2^ — s in the
equation 5^ + ^" — 2A5# + ^Jc^ = ; so that we have for the equation of the
14- A
curve referred to a focus and the centre, — ~- s^ + #^ — (1 + A) «* + ^ = ;
from which we obtain
L+ *a2* ^r''—{l + h)at + Tc"^' zz 0, or r" + Ar^p' = 4"^ (T (2t — a) .
Now T^ + 1&^ is the square of the distance from the centre to the foot of the
perpendicular dropped from the focus, and a (2t — <t) is the product of the
perpendiculars from the two foci, which is constant and equal to V' : hence the
distance from the centre to the foot of the perpendicular dropped from the
focus upon any tangent is constant and equal to \ — ^— b^ = \- V^ -zz a\
2 1 — e^
i. e., the feet of all these perpendiculars are on a circle described upon the
major axis as diameter.
Tlie Conic referred to the Extremities of a Biameter. — Taking up the
general equation of a conic referred to conjugate basics, — + — zz Z^, and
(j a
making the general substitution for a motion of the initials along the base,
s= , r== (v.iiere % is the distance between the new
initials), we obtain, dropping the accents of the variables,
Franklin, Bvpunctual Coordinates, 173
If the extremities of the diameter are taken as the new initials, we have
(p. 160) -^ -f _ = , --- + — = ; also, (since m, m', are the roots of the
& or & or
equation {& + d'^)^!? + 2c^/|U + cH'^ == 0, and w, ti', the roots of the equation
^2/2 ^72
{& + rf') ^' — "^dHv + dH'' = 0) mm' = v^^ » ^^' = -^^^, i so that the
r + « cr -i- d^
equation of the curve referred to the extremities of a diameter is
4:St / b^\
— -2 -r-72 = ^^ = 4a^ ; or ( since c^ + d"^ = 5- ) , 8t=W]
that is, the product of the distances cut off by any tangent on the tangents at
the extremities of a diameter is equal to the square of half the conjugate
diameter.
The equation of a conic in point-coordinates has for its coefficients the
minors of the determinant of the coefficients of its equation in line-coordinates;
hence the equation in point-coordinates of the conic referred to the extremities
of a diameter is b^pq — — zz , or pq = —^. Representing, then, by 7t, x,
the negative reciprocals of p^ q, we have nx zz 46^ ; that is, the lines joining
any point of the conic with the extremities of a diameter cut off, on the
tangents at those extremities, distances whose product is equal to the square
of the conjugate diameter.
44
DESIDERATA AND SUGGESTIONS.
. By Professor Cayley, Cambridge^ England.
No. 2. — THE THEORY OF GROUPS: GRAPHICAL REPRESENTATION.
In regard to a substitution-group of the order n upon the same number
of letters, I omitted to mention the important theorem that every substitution
is regular (that is, either cyclical or composed of a number of cycles each of
them of the same order). Thus in the group of 6 given in No. 1, writing
a, ft, 6*, rf, e^ f in place of 1, a, /?, y, S, e, the substitutions of the group are
1, ace.bfd^ aec,hdf\ ab.cd.ef^ ad,be,cf^ af.bc.de.
Let the letters be represented by points ; a change a into b will be repre-
sented by a directed line (line with an arrow) joining the two points ; and
therefore a cycle abc^ that is, a into 6, b into c, c into a, by the three sides of
the trilateral abc^ with the three arrows pointing accordingly, and similarly
for the cycles abcd^ &c. : the cycle ab means a into 6, b into a, and we hav^e
here the line ab with a two headed arrow pointing both ways ; such a line
may be regarded as a bilateral. A substitution is thus represented by a
multilateral or system of multilaterals, each side with its arrow ; and in the
case of a regular substitution the multilaterals (if more than one) have each
of them the same number of sides. To represent two or more substitutions
we require different colours, the multilaterals belonging to any one substitu-
tion being of the same colour. •
In order to represent a group we need to represent only independent
substitutions thereof; that is, substitutions such that no one of them can be
obtained from the others by compounding them together in any manner. I
take as an example a group of the order 12 upon 12 letters, where the num-
ber of independent substitutions is zz 2. See the diagram, wherein the
continuous lines represent black lines, and the broken lines, red lines.
171
Oayley, Desiderata and Suggestions.
175
The diagram is drawn, in the first instance, with the arrows but without
the letters, which are then affixed at pleasure ; viz: the form of group is quite
independent of the way in which this is done, though the group itself is of
course dependent upon it. The diagram shows two substitutions, each of
them of the third order, one represented by the black triangles, and the other
by the red triangles. It will be observed that there is from each point of the
diagram (that is, in the direction of the arrow) one and only one black line,
and one and only one red line; hence, a symbol, J?, "move along a black
line," ^, move successively along two black lines," BR (read always from
right to left), " move first along a red line and then along a black line," has
in every case a perfectly definite meaning and determines the path when the
initial point is given ; any such symbol may be spoken of as a '' route."
The diagi'am has a remarkable property, in virtue whereof it in fact repre-
sent a group. It may be seen that any rqjite leading from some one point a
to itself, leads also from every other point to itself, or say from b to 6, from c
to (?,... . and from I to L We hence see that a route applied in succession to
the whole series of initial points or letters abcdefgkijklj gives a new arrange-
ment of these letters, wherein no one of them occupies its original place ; a
route is thus, in eflfect, a substitution. Moreover, we may regard as distinct
routes, those which lead from a to a, to ft, to c, . . . to Z, respectively. We
have thus 12 substitutions (the first of them, which leaves the arrangement
unaltered, being the substitution unity), and these 12 substitutions form a
group. I omit the details of the proof; it will be sufficient to give the square
obtained by means of the several routes, or substitutions, performed upon
the primitive arrangement abcdefgkijklj and the cyclical expressions of the
a b
c
d
e
f
9 h
m
%
m
3
k
I !
1
b c
1
a
e
f
d
h i
9
k
I
3 '
abc . def .
ghi . jkl (
c a
b
f
d
e
» 9
h
I
•
3
k
acb . dfe .
gih . jlk
d I
A
a
9
•
3
e c
k
f
•
b
ad , bl . ch .
eg .fj .ik
« J
•
b
h
k
f «
I
d
9
c
aeh . bjd .
cil . fkg
f ^
9
c
•
*
I
d b
•
3
e
h
a
of I . bkh .
cgd . eij
9 f
k
I
c
m
3 d
b
a
e
h
agj . bfi .
eke . dlh
h d
I
•
3
a
9
k e
c
b
f
•
ahe . bdj .
cli . fgk
i e
•
k
b
h
I f
a
G
d
9
ai . be , cj .
dk . fh . gl
. «
e
h
k
b
a I
f
9
c
d
ajg . bif .
cek . dfd (:
k g
f
•
I
I
c
b 3
d
h
a
e
ak , bg . cf .
di , el , hj
I h
d
9
•
J
a
c k
e
•
^
b
f
alf . bhk .
•
cdg . ^i
= B)
:=i?).
176 Cayley, Desiderata and Stiggestions.
substitutions themselves : it will be observed that the substitutions are unity,
3 substitutions of the order (or index) 2, and 8 substitutions of the order
(or index) 3.
It may be remarked that the group of 12 is really the group of the 12
positive substitutions upon 4 letters abed, viz., these are l^abc, acb , abd, adbj
acd, ado, bed, bdc , ab.cd, ac.bd, ad. be.
Cambridok, 16th May^ 1878.
ON THE ELASTIC POTENTIAL OF CRYSTALS.
By William E. Story.
The theory of elasticity in its applications to homogeneous isotropic
bodies has been more or less completely developed by Poisson, Cauchy,
Lame and Clebsch, while Riemann's " Differential-gleichungen " contains an
admirable abstract of it.
On a particle of a solid body may act three kinds of forces :
1) External forces^ attractions and repulsions by external masses;
2) Surface-forces^ pressures or tensions applied directly to elements of
the surface;
3) Elastic forces^ molecular forces due to the influence of neighboring
particles of the body itself.
Elastic forces are produced by unequal displacements of the particles,
causing a change in their relative positions, and hence a change in the mole-
cular forces acting between them. Surface-forces aflfect the internal particles
of the body only indirectly, by causing a displacement of the particles of the
surface, giving rise to elastic forces which spread through the interior. It is
the elastic forces alone that I shall consider in this paper. These may be
considered as forces tending to prevent any change in the relative positions of
the particles on one side of a section (plane or curved) of the body with
respect to those on the other side ; and, because the mutual action of the par-
ticles becomes inappreciable when the distance between them exceeds a very
small limit, a particle on one side of the section will be affected only by those
particles on the other side which lie very near the section and within a small
distance of the particle in question. So that the whole mutual effect of the
particles on the two sides of the section may be treated as the sum of the
mutual effects of particles on the two sides of the very small portions (which
may be considered plane-elements) of the section ; and it is only necessary to
consider the elastic forces on such plane-elements, which vary from element to
element of the section, but, in the immediate vicinity of any given point of the
section, are proportional to the plane-elements on which they act. It is there-
46 m
178 Story, On the Elastic Potential of Crystals.
fore convenient to call the components of the elastic force acting on any plane-
element, the ratios of the actual components to the area of the element. The
components of the elastic force acting on any plane-element passing through
a given point may be expressed in terms of the components of the elastic
forces acting on any three mutually perpendicular plane-elements through the
point, e. g. three elements parallel to the planes of a rectangular system of
coordinates.
Let cr, y, z be the coordinates of any point of the body referred to a fixed
rectangular system, and let three plane-elements of areas e,,, Sy, f^, whose nor-
mals are parallel to the axes of a-, ;y, z respectively, pass through this point ;
then in notation of Cauchy the components of the elastic forces acting on
€„ fy, f, in the directions of the three axes will bo X^, Y^^ Z, ; JTy, Yy^ Zy\
X., 1"^, Z^ : in each of which the large letter denotes the direction of the com-
ponent, and the suffix the direction of the normal to the plane-element on
which it acts. Let, further, the density at the point x^ y, z be p.
If ,r, y, z is a point in the interior of the body, let the components of
the resultant of the external forces acting on a very small mass m at this
point be mX^ niY, mZ ; then the conditions for the equilibrium of this small
mass are
(1)
9X +
hx
81/
+
hX,
hz ~
:0,
9Y+
hx
8Y,
+
hz~
0,
pZ +
8x
hZy
^y
+
hZ,
hz ~
0,
Yg Zy , Z^ X^y Xy Yg .
If, however, or, y, 2 is a point of the surface, let a, /?, y bo the direction-
cosines of the inner normal to the surface, and eH, ell^ eZ the components of
the surface-force on an element b of the surface ; then the conditions for the
equilibrium of an infinitesimal solid element immediately under the surface-
element e are
r E +aX^ + pXy+yX, = 0,
(2) \ H+aY^ + (3Yy + yY, = 0,
( Z+aZ^ + (3Zy + yZ^ = 0,
where the surface-force is considered positive if a pressure and negative if a
tension.
Story, On the Elastic Potential of Crystals. 179
It is well known that, if t/, t;, w are the displacements, in the directions of
the axes, of the point x^ y, z, and if
(3)
hu hv hw
hv hw _ _ hw hu _ _ hu hv _ '
^ + ^ -^'- ^^ ^ "^ '^2 -'"'-"'" ^ "^ ^x - ''«' -^"
then there exists a function <l> of x,., y^, z^, y,, z,, Xy, such that
(4)
hx^ hfy h,
V — 7 —^^- 7 — Y —^ Y — V — ^'^
oyg hz, "^ hxy
This function 4>, which may be called the elastic potential^ is homogeneous
of the second degree in x^y ^y, 2„ y^, z^, Xy, and contains therefore in general 21
terms, whose coefficients are characteristic for the body under consideration,
depending only on its structure in diflferent directions. For a homogeneous
body, i. e. one, every particle of which is similarly surrounded by particles,
the coefficients of 4> are constant, but for a non-homogeneous body they diflfer
from point to point, i. e, are functions of the coordinates x, y, z. For a homo-
geneous isotropic body the form of 4> has been determined to be
(5) ^ = 1-^ {x. +yy + Zgy + (i {xl +yl+zl)+^fi if; + zl + x^) ,
where ^ and (i are constants. I now propose to determine its form for homo-
neous crystalline bodies.
Each crystalline system has a form of symmetry with respect to three
mutually perpendicular axes which is not altered by a transformation to any
one of certain other systems of mutually perpendicular axes having the same
origin. We may then take one of these systems of axes as the basis of a
system of coordinates x, y, z, and any other of them as the basis of a system
of coordinat.es x", y, zf. The external form of symmetry of a crystal must, it
would seem, be also a form of symmetry of arrangement of its particles ; if
this be so, a transformation of coordinates from x, y, z to .r', y, z^ must leave
the coefficients unchanged. This will be the basis of the following determina-
tions of the relations between the coeflBcients, on which relations the form of
O depends.
The transformation from x^ y, z to x^^ y, 2' can be effected by a combina-
tion of two simple kinds of transformations, viz. by inversions of axes and
180 Story, On the Elastic Potential of Crystals.
rotations about axes through certain angles. There are six such transforma-
tions, which I will designate by the first six letters of the alphabet, as follows :
a) inversion of the axis of x ,
3(f z=z iT, ^ =1 y ^ Z' =1 Z ^ U = 1^', |? = i/, W = V/ J
a\ = a^ , yy =^ tf^ , 2^ = 2j'^ , y^, = y'z' j ^x = — ;• ^x' » ^y = — ^y' J
h) inversion of the axis of y,
^r=a',y = — y, z! =1 Z ^ U z= U\ V = — 1/, w = w' J
^x = «^ z' J j^y = !/y' J ^z ^==- ^z* •i yz ^=^ y y > ^x = ^«' > ^y = ^y' ;
c) inversion of the axis of z,
•^x = '^ x"i yy = ^y' ) ^z =^ ^z' ^ yz "= y^' > ^x = ^x' ) ^y *= ^y' I
rf) rotation about the axis of x through the angle a ,
or' = a; , ;/ = y cos a + z sin a , z' = — y sin a + ^ cos a ,
u =^ u' ^ V = if cos a — w' sin a , w = if sin a + w' cos a ,
a^^ = x"^ , y^ = y'y, cos^ a — y^ sin a cos a + 2^^ sin^ a ,
z, = }f^ sin^ a + y^ sin a cos a + ^*' cos^ a ,
^r = (/y — ^'?j sin 2a + y^ cos 2a ,
2?^ = sf^ sin a + /^ cos a , ^y = ar'y. cos a — 2;^^ sin a ;
e) rotation about the axis of y through the angle /?, by formulae found from
those of d) by changing x to y^y to z^ z to x^ and a to /? ;
/) rotation about the axis of z through the angle y, by formulae found from d)
by changing x to z^ y U> x^ zio y^ and a to y.
The general form of 4> is
O ZZ «n + ^ + f^Z2jfij + (h^A + 2a23yy^x + 2^13^^ + 2tfi2^,3^y
+ 2ai4^xy. + 2a,5^x2x + 2ai6^x^y + 2a24y«yx + "^(h^^z
+ 2a263^y^y + "^d^^fizHz + 203525^ + 2a3e2^y + ^44^^
+ ^552:2 _j_ ^^^,2 ^ 2a562;^y + 2(Z46^j^. + 2^4^^ •
That the transformation a) shall leave O unaltered in form it is necessary
and sufficient that
(7) a,5 = ^2.-, =1 Oas = ^45 = «56 = ^16 = ^^26 = Afge = «46 = 0-
That the transformation J) shall leave the form of O unaltered it is necessary
and sufficient that
(8) (^14 zz. tti^ zz: a^ zz. a^ ":=. OiQ zz a<^ "zz a^ 1=. a^ zz 0,
That the transformation c) shall leave the form of ^ unaltered it is necessary
and sufficient that
(9) ai4 = 024 = 034 = 046 = dig = 026 = flfsB = ^56 = 0.
(6)
Story, On the Elastic Potential of Crystals. 181
That the transformation d) shall leave the form of O unaltered it is necessary
and sufficient that
(^12 — a,3) sin a = , (^55 — a^) sin a = , {a^ — 033) sin a = ,
ai4 sin a = , ag^ sin a = , o^g (1 — cos a) = , a^^{l — cos a) = ,
(^22 + ^33 — 2^23 — 4a44) sin a cos a = ,
(^4 + <h^ sin a = , (a^,4 — 034) sin a cos a = ,
(«25 + ^35) (1 — cos a) = , (a,6 + «3e) (1 — cos a) = ,
[(«26 — ^) + (^ — «36)] sin a cos a + (cos a — cos 2a — sin a) ^45
(10) \ + (cos a — cos 2a + sin a) a^/^ = ,
[(«25 — ^) — («26 — ^se)] sin a cos a + (cos a — cos 2a + sin a) a^
— (cos a — cos 2a — sin a) ^45 z= ,
(<3525 — ^0 (cos a — cos 2a — sin a) + [a^^ — a^) (cos a — cos 2a + sin a)
— 4 (^45 + ^46) sin a cos a = ,
(«25 — ^) (cos a — cos 2a + sin a) — {a^ — a^ (cos a — cos 2a — sin a)
— 4 {a^ — a^ sin a cos a = .
The conditions (10) are, of course, satisfied by a = 0. They will also be
satisfied by a multiple of 90° when certain relations hold between the
coefficients, as follows :
if a = 90° or — 90°,
r^ ^ V J ^4 ^^ ^6 ^^ ^16 I^ ^ 13 ^26 ^^ ^ ^^ Oy^ 13 a45 HZ ^46 13 Ose ZZ U ,
( ^2 ^^ ^13 ) ^ ^^ ^ J ^56 ^ ^ > ^ ^^ ^ 5
if a = 180°,
(12) dis ZZ ^25 = ^ = «45 = <3^16 = ^ = ^ = ^46 = ,
which are identical with (7), so that, if an axis can be inverted without pro-
ducing any effect on 4>, a rotation about that axis through an angle of 180°
will also produce no effect, and vice versa j as is also evident from the fact that
all three axes may be inverted without producing any effect on the most gene-
ral form (6) of <I>.
If a has neither of the above values.
(13)
I «14 = <l24 = ^34 = «15 =
( ^12 ^^ ^13 > ^22 ^^ ^33 ^^ ^5
^14 = <l24 = ^34 = «15 ZZ ai6 = ©56 = ,
^23 "T ^^44 J ^66 ^^ ^66 ) ^ ^^ ^86 ? ^26 ^^ ^36 >
and either
(13.)
^26 =
^ =
= «46 =
^
=^^36=
«46
-0,
or
(13,) 4cos^
a
+ 4
cosla -
-2
cos^ a -
-2
cos' a -
-4
cos^a
4 cos a — 1=0
46
182 Story, On the Elastic Potimtial of Crystals.
(whose only real roots between — 1 and + 1 are cos a ■=. — 0.607 + and
cos a = — 0.389 — ), i e. if a rotation about an axis through any other angle
than 90^ 180^ 270° (or — 90°), cos ^-^^(— 0.607 +) or cos t" ^J (— 0.389 —)
does not affect the form of 4>, a rotation about this axis through any angle
whatever will not affect it, i. e. there exists perfect symmetry about this angle.
The transformations e) and /') give conditions similar to (10), (11), (12)
and (13), found from them by a cyclic interchange in each set of suffixes 1,
2, 3 and 4, 5, 6 and in a, ^, y.
Each of the crystalline systems admits of a characteristic combination of
the above transformations.
The iriclinic system admits only of an inversion of all the axes at the
same time, which leaves the most general form of 4> unclianged without any
conditions between the coefficients, /. e, (6) is the general form of O for tri-
clinic crystals.
The monoclinic system, in which the axis of x is taken in that axis of the
crystal which is perpendicular to the plane of the two others, allows only the
transformation a) and d) in which a zz 180°, i. e. its conditions are (7) and
(12), which are identical, and
(14) } + 2ai2^xj^y + 2auX,i/, + 2a^^f/^i/, + 2a^z,i/, + a^f.
The trimetric system, in which the axes of »r, y and z coincide with the
axes of the crystal, admits of each of the transformations a), b) and c), hence
for this system the conditions are (7), (8) and (9), and the form of O is
- g. J * = «ii^ + a^ify + a^^^l + 2a^i^^, + 2a,:,x^, + 2a,2X^y
1 = «44j^ + d'^zl + «6c4-
The dimeiric system, whose principal axis is the axis of x and whose
secondary axes are those of y and 2, admits of the transformations a), ft), c)
and d) where a = 90°, so that the conditions are (7), (8), (9) and (11), and the
form of 4> is
.g. I * = «ii^ + ^22 (^ + zi) + 2a2!&yZ, + 2^120:^ (^y + z,)
\ +auyt + Oo5 (2Jx + 4).
The monometric system, whose axes are the axes of x^ y and z, admits of
the transformations a), ft), c), rf), e) and/) for a = 90°, ^ = 90°, y = 90° ; so
that the conditions are (7), (8), (9), (11) and, similar to (11),
Story, On the Elastic Potentkd of Gystals. 183
^14 = ^IG ^^ ^24 ^^ ^25 ^^ ^26 ^^ ^34 ^^ ^3(1 ^^ ^.'Wi ^^ ^40 ^^ ^45 ^^ ^ )
©23 ^^ ^12 J ^11 ^^ ^33 ? ^44 ^^ ^66 > ^15 ^^ ^3r> )
tfl4 ZZ 0,5 ZZ fl?24 ^^ ^25 ^^ ^34 ^^ ^35 ^^ ^:J6 ^^ ^56 ^^ ^46 ^^ ^45 ^^ ^ j
^23 ^ ^13 > ^11 ^^ ^22 ? ^44 ^^ ^ix> ) ^16 ^^ ^26 J
and the form of 4> is
(17) O zz a,! {rl + y^ + 2?) + 2^12 (y^ + ^.o;, + x,^,) + ^44 (j^; + z] + 4).
The hexagonal^ as well as a«y regular prismatic system^ other than one
whose secondary axes include an angle of 90° (because the conditions (13,)
omitted in case a = cos^~^^ ( — 0.607 +) or a = cos^~^^ ( — 0.389 — ) are more
than replaced by (7) ), whose principal axis is that of or, admits of the trans-
formations a)^ i), c), d) ; the conditions are then (7), (8), (9), (13) and (13,)
and 4> has the form
f * = «n^x + dn {fy + z]) + 2a,,^yyZ, + 2^^120;, {y,j + z,)
1 + ^ («22 — ^23)^ + «5.5 {A + xl) .
It is to be remarked that (16) differs from (18) only in the coefficient
of ^, which is independent of the other coefficients in the former case, but
dependent on the coefficients of yl + z] and yy z^ in the latter case, so that, as
far as their ehastic relations are concerned, square prismatic crystals seem to
be the most complex of regular prismatic forms, in as much as they contain
one more constant than any other.
From the above determinations of 4> it will be seen that the elastic poten-
tials of isotropic bodies, of monometric, hexagonal, dimetric, trimetric, mono-
clinic and triclinic crystals contain respectively 2, 3, 5, 6, 9, 13 and 21 con-
stants. The forms of 4>, (5), (17), (18), (16), (15,) (14) and (6) substituted
in (4) give the forms of the components of the elastic force in these differ-
ent cases of homogeneity, and these components substituted in (1) and
(2) give the conditions for equilibrium under a given set of external forces
X, F, Z and surface-forces H, //, Z. The conditions for motion, e. g, elastic
vibrations, will be found from those for equilibrium by the substitution
d^ii d^v d^w
oi X — -— , Y — i^j-^ — ^^- foJ* ^? Y^ Z respectively, t denoting time.
d?U y
df
d?v „
df'"^
d^w J.
-df ^''
THfiORIE DES FONCTIONS NUMfiRIQUES SIMPLEMENT
PfiRIODIQUES.
Par Edouard Lucas, Professeur au Ijycee Charlemagne^ Paris.
Oe memoire a pour objet I'etude des fonctions symetriques des racines
d'une equation du second degre, et son application a la theorie des nombres
premiers. Nous indiquons des le commencement, I'analogie complete de ces
fonctions symetriques avec les fonctions circulaires et hyperboliques ; nous
montrons ensuite la liaison qui existe entre ces fonctions symetriques et les
theories des determinants, des combinaisons, des fractions continues, de la
divisibilite, des diviseurs quadratiques, des radicaux continus, de la division
de la circonference, de Tanalyse indeterminee du second degre, des residus
quadratiques, de la decomposition des grands nombres en facte urs premiers,
etc. Cotte methode est le point de depart d'une etude plus complete, des pro-
prietes des fonctions symetriques des racines d'une equation algebrique, de
degre quelconque, a coefficients commensurables, dans leurs rapports avec les
theories des fonctions elliptiques et abeliennes, des residus potentiels, et de
I'analyse indeterminee des degres superieurs.
Section I.
Definition d^s fonctions numMques simpleinent pSriodiques.
Designons par a et ft les deux racines de Tequation
(1) :x? = Px—Q,
dont les coefficients P et Q sont des nombres entiers, positifs ou negatifs, et
premiers entre eux. On a
a + b = P, ' ab=Q]
et, en designant par 5 la difference a — b des racines, et par A le carre de
cette difference, on a encore
184
Lucas, TMorie d^s Fonctions Numerigues Simjjleiyient PSriodigues. 185
Cela pose, nous considererons les deux fonctions numeriques U Qt Fdefinies
par les egalites
(2) J7„ = ^-^^\ F„zza«+A".
a —
Cos fonctions Un et V^ donnent naissance, pour toutes les valeurs entieres
et positives de n, a trois series d'especes differentes, selon la nature des racines
tf et 6 de Tequation (1). Cette equation peut avoir :
1°. Les racines reelles et entieres ;
2°. Les racines reelles et incommensurables ;
3^. Les racines imaginaires.
Les fonctiom numeriques de premiere espece correspondent a toutes les
valeurs entieres de a et de J, et peuvent etre calculees directement, pour
toutes les valeurs entieres et positives de n, par Temploi des formules (2).
Si Ton suppose plus particulierement a zz 2 et 6 = 1, on trouve, en formant
les valeurs de U^ et de F„, les series recurrentes
n: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ,
U^: 0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, ,
F„: 2, 3, 5, 9, 17, 33, 65,-129, 257, 513, 1025, 2049,
6tudiees pour la premiere fois par Tillustre Fermat. Nous observerons,
des maintenant, que la serie des V^ est contenue, pour les trois cas que nous
considerons, dans la serie des U^^ puisque les formules (2) nous donnent la
relation generale
(3) U^=U„V,.
Les fonctions numSriques de seconde espece correspondent a toutes les var
leurs incommensurables de « et de J dont la somme et le produit sont
commensurables. On peut les calculer en fonction de la somme P et du
discriminant A do Tequation proposee, au moyen des formules suivantes. Le
developpement du binome nous donne
2-0-= /»"+ % P»-«S+ -^"-Z-il P" - ^^ + ^ (" ~ \)/" ~ ^ ) P— »5» + . . .+5-,
1 1.2 1 . 2 . o
2-A* = P"— f P— ^ + <'^-}) pn-2g2_ «(«— 1)(^— ^) p«-35»_, ^ .+(—*)•;
1 1*I2 1.2.0
et, par soustraction et par addition,
47
2»
186 Lucas, Theorie des Fonotiom Numeriques Simplemeni FSriodigues.
"~ 1 "^ 1.2.3
(4) ^ + n(n-l)jn-|(»-3)(n-4) p„_,^, _^ ; ^
On obtient ainsi, pour les premiers termes,
fToZzO, U, = l, L\ = P. U,= P'—Q, U, = F' — 2PQ,
Fo=2, V,= F, V,= P' — 2Q. Vs = F' — 3PQ, V, = P' — 4P'Q + '2(^.
Les fonctions numeriques de seconde espece les plus simples correspondent
aux hypotheses
P=:l, Q = — h A = 5,
ou a I'equation x^ = x + 1;
on a, dans ce cas,
et, par suite, en designant par m„ et v„ les fonctions qui en resultent,
_ (1 + V 5)" - (1 — V 5)" ; _(1 + V5)'-+(l — V5)"
2"V5 ^'^
On forme ainsi, pour les premieres valeurs de n entieres et positives, les
series
n: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ,
u, : 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ,
t), : 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199,
La serie des w« a ete eonsideree pour la premiere fois par Leonard Fibonacci,
de Pise.* Elle a ete etudiee par. Albert GriRARD,f qui a observe que les
trois nombres w„, u^, u^ ^ i ferment un triangle isoscele dont Tangle au sommet
est a fort peu pres egal a Tangle du pentagone regulier. Robert Simson J a
* II liber Abbaci di Leonardo PUano, pubblicaio seeondo la Uzimie del Codiee MagliabechianOy da B. BoKCOM-
PAONi. Roma, 1867. Piig. 283 et 284.
f L^ariihmeiique de Simon Stkvin, de Bruffea^ revue^ corrigee et augmentee de plusieura traictez ei annotations
par Albert Girard, otc. Leide, 1633. Pag. 169 et 170.
J PhUoaophieal Transactions of the Royal Society of London, Vol. xlviii, Part I, for the year 1768. An
explication of an obscure passage in Albert Girard's Commentary upon Simon Stevin's Works. Pag 368
et suiv.
Lucas, Tkeorie. des FoncUons Num^riquea Simplement Phiodigues. 187
fait remarquer en 1753, qire cette serie eat donnee par le calcul des quotients
et des fractions convergentea des expressions irrationnelles
V^+ 1 , VS — 1
En 1843, J. BiHET* donne, au inoyert de cette aerie, I'expreasion du denom-
brement des combinaisona discontigues. En 1844, LAMEf indique I'appli-
cation que Ton peut faire de cette aerie a la determination d'une limite
auperieure du nombre des operations a faire dana la recherche du plus grand
coraraun diviseur de deux nombrea entiers.
Nous prendrons ausai quelquefois pour exemple la serie U„ de aeconde
eapece, donnee par lea hypotheaea
P=3, Q = —\, A = 2^.2,
ou par I'equation
9, 10, 11, . . ;
985, 2378, 5741, . . .
2786, 6726, 16238, . . .
que noua designerona aoua le nom de SIibies db Pell, en I'bonneur du
mathematicien de ce nom qui resolut, le premier, un celebre probleme
d'analyae indeterminee propose par Fbrmat, et concernant la resolution
en nombres entiers, de I'equation indeterminee
j:^-A/ = ±1.
Lea fonctUms num^riques de troisieme espece correspondent a toutes les
valeurs imaginaires de « et de A dont la aomme ot le produit sont reela et
commenaurables. Les plus aimplea proviennent des hypotheses
F~l, Q=l, A = — 3;
on a, dans ce cas,
1+ V— 3 , 1— V— 3
«=^^X— ' * = 2—'
par consequent a et J aont lea racines cubiques imaginaires de I'unite negative ;
de plus,
f4. = 0, t7^ + , = {— 1)", U^^t= (—1)".
* Ck-mplei rendui de t'eeadimU dei acitncra lie Pnrii, tome, xvii, pag 6(i2 ; tome nix, pag. OSS.
t Oamptet rendua, etc., toma lix, pag. 8ST.
Oq a alors les series
«: 0, 1, 2, 3,
4,
6,
6, 7,
8,
If.- 0, 1, 2, 5,
12,
29,
70, 169,
408,
K: 2, 2, 6, 14,
34,
82,
198, 478,
1154,
188 Lucas, TMorie des Fonctions Numerigues Sinvplenient Periodiques.
Ainsi les valeurs de ?7„ reviennent periodiquement dans Tordre
0, 1, 1, 0, -1, -1,
et donnent lieu a un grand nombre de formules simples deduites des pro-
prietes generales des fonctions U^ et F^, et concernant la trisection de la
circonference.
Quelquefois aussi nous considererons les series analogues deduites de
Tequation
^ = 2a: — 2,
dans laquelle
a-l + V^=^, ftzzl — V^^, A = — 2^,
et les series deduites de I'equation
x'-2x — ^,
dans laquelle
« = 1 + V^=^, Ai^l — V'^^, A = —2X2^;
nous designerons les series obtenues dans cette derniere hypothese, sous le
nom de series conjuguees de Pell.
Section II. .
Des relations des fonctions U^ et F„ avec les foncUons circulaires et hyperboliques.
Si Ton fait
dans les formules
z = — Log. nep. j ,
cos {ZW — 1) jr
e-'
sin (z V — 1) =
on obtient
2V — 1
in /""V— i T «N 1 r«' *n
Lucas, Theorie des Fonctions Num4riques Simpleraent Periodiques, 189
on a done, entre les fonctions U^ et V^^ et les fonctions circulaires, les deux
relations
(5)
F„ = 2q" cos (^-^ Log. 1) ,
n
II resulte immediatement de ce rapprochement que chacune des formules de
la trigonometrie rectiligne conduit a des formules analogues pour U^ et F^, et
inversement.
Ainsi la formule (3)
correspond a la formule
sin 2z = 2 sin z cos z ;
les equations
(6) V^ + hU, = 2a\ V,-hU, = 2b\
que Ton deduit immediatement des formules (2) correspondent exactement
aux relations
cos z + V — 1 sin 2 zz e'^- ^ , cos 2; — V — 1 sin z = e"'^'^ ,
et les formules (4) sont entierement analogues a celles qui ont ete donnees
dans les Actes de Leipzick^ en 1701, par Jean Bernoulli, pour le developpe-
ment de — ; et de cos nz suivant les puissances du sinus et du cosinus de
sin z
Tare z. Ainsi encore les formules
[F„ + 3 J7„][F„ + 5?7-„] = 2[F„ + , + hU^^„-\ ,
que Ton deduit des relations (6) coincident avec les formules
(7) I
(cos .r + \/ — 1 sin or) (cos y + V — 1 sin ^) ir cos {x -\- y) + V — 1 sin {x + i/)j
(cos a; + V — l.sin xy zz cos ra: + V — 1 sin rx^
qui ont ete donnees par Moivbe.
Nous ferons encore observer que si, dans Teq nation (1), on pose
X=x', azza^ /3zzJ%
les quantites a et ^3 sont les racines de Teq nation
(8) X'' = VrX — Q' .
48
190 Lucas, Theorie des Forwtions Nwiieriques Simpleinent Periodigues,
Par consequent, chacune des formules qui appartiennent a la theorie presente
peut etre generalisee, en y rempla^ant U^ et V^ par -=y ^^ ^nn P P^r F^, Q par
Q% et la difference 8 des racines a et 6 par la difference h U^ des racines a et /?
de Tequation (8).
Les formules (4) deviennent ainsi
Ur 1 "^ 1.2.3
(9)
n{n — l ) {n — 2) (n — 3) {n — 4) ^ 774 t,-,. - 5 ,
"^ '^1.2:3.4.5 ' ' "^ ■•
-/
2" - * r,, = r;' + " ^" ^ ^ a f/,* v; - ^
« (w — 1) (« — 2)(n — 3) J „, ^,_,
+ 172-3T4 "^ ^' ^' + • • •
D'ailleurs, nous laisserons de cote, pour Tinstant, les autres precedes de trans-
foVmation de Tequatiou (1) par substitution de variable, ainsi que Tetude des
fonctions plus general es
AU,+ Bv^ + a
dans lesquelles A, B, C designent des nombres entiers quelconques, positifs
ou negatifs.
Section III.
Des relation.^ d^ recurrence pour le calcul des valeurs des fonctions ?7„ et V^.
he calcul des valeurs de Un et de F„ qui correspondent aux valeurs
entieres et consecutives de n, s'effectue rapidement au moyen de formules
entierement analogues a celles de Thomas Simpson :
sin (n -{- 2) z zz 2 cos z sin (n + 1) 2 — sin nz ,
cos {71 + 2) ;25 = 2 cos z cos {n + 1) z — cos nz .
En effet, multiplions par x^ les deux membres de Tequation (1), et rempla^ons
successivement x par a et 6, nous obtenons
a'^^^ = Pa"" ' — Qa\ A" ^^ = Pb""'' — QJ^
et, par soustraction et par addition,
U,,,= PU^^,-QUr.,
(10) {
(11) {
Lucas, Theorie des Fonctions Nunienques Simplenient Periodiques. 191
Ces formules nous font voir que les fonctions U ei Fforment, pour les valeurs
entieres et consecutives de n, deux series recurrentes de nombres entiers. Ces
series ont la meine loi de formation, mais elles different par les conditions
initiates. Nous generaliserons ces formules par I'emploi du calcul sym-
bolique. En eifet, en designant par F une fonction quelconque, on tire
evidemment de I'equation (1)
si Ton remplace x par a et ft, on a
a^'F {a') = a^'F {Pa — Q) , b^F (ft^) z= b^'F {Ph— Q) ,
et, par soustraction et par addition, on obtient les egalites symboliques
U^'F {U'') = U'^F {PV — Q) ,
V''F{V')=V''F{PV— Q),
dans lesquelles on remplace, apres le developpement, les exposants de U et de
Fpar des indices, en tenant compte de I'exposant zero. Ainsi les symboles
U^ et PU — Q, V^ et PV — Q sent respectivement equivalents, et peuvent
etre remplaces Tun par Tautre dans les transformations algebriques.
On a, par exemple, dans la serie de Fibonacci, les resultats suivants
u''-p = ur{u — ly,
qui sont eritierement analogues a ceux que Ton pent obtenir dans la theorie
des combinaisons ou du triangle arithraetique, et, en particulier dans la
formule du binome des factorielles, due a Vandermonde.
En prenant, pour point de depart, I'equation
x^ =: X — 1 ,
on trouvera encore de nouvelles relations entre les coefficients de la meme
puissance du binome.
La consideration de I'equation (8) conduit aux relations suivantes
f^n + 2r ^ f^r ^ n -\- r ^ ^ n i
^ n + 2r ^^^ ' r ^ n -\- r V ' n y
qui perraettent de calculer les valeurs des fonctions U^ et V^ qui correspondent
i des valeurs de I'argument n en progression arithmetique de raison r.
Inversement, on trouvera, dans la theorie des fonctions circulaires et
hyperboliques, des formules analogues aux formules (11) et (13).
(12) {
(13) I
192 Lucas, Theorie des Fonctiam Nuvfieriqms Simplement Feriodiqties.
Section IV.
Des relations des fonctiom U^ et \\ avec les determinants.
On peut exprimer Un et U„r, F„ et V„r au moyen de determinants; en
eifet, on a les formules
U, — PU,+ QU, = 0,
U, — PU,+ QU,
U,-PU, + QU,
*
* =0,
* =0,
• •
• •
• • •
• •
» • • •
• •
c^»+i- PU„-\- qu^.
• •
- 1
4b *
-0;
on en deduit
-p,
+ 1,
0,
0, . . . .
[n colonnes),
+ Q,
-p,
+ 1,
0, . . . .
(14) f^„ , , = (- 1)-
0,
+ «,
-P,
+1, ....
•
0,
0,
+ Q.
-p, ....
On obtient aussi
-p.
+ 2,
0,
0,
(71 colon nes),
•
-P,
+ 1,
0, . . . .
(15) F, = (- 1)» 0,
+ Q,
-P,
+ 1, . . . .
•
•
0,
0,
• • • •
+ «,
-p,....
•
• •••••
On verifie les resultats que nous venons de trouver, en developpant les deter-
minants suivant les elements de la derniere ligne ou de la derniere colonne.
Les valours de -~ et de Fn^ s'obtiennent encore au moyen des determinants,
en rempla§ant comme a I'ordinaire P par F^. et Q par Q".
Enfin, nous ferons observer que ces formules sont susceptibles d'une
grande generalisation ; en eifet, dans les formules (11) qui contiennent une
fonction arbitraire, faisons n successivement egal a 1, 2, 3, .... m ; nous
obtenons alors m equations desquelles on tirera la valeur de Tune ou Tautre
des fonctions U et F.
Lucas, Theorie des Fonctions Num^iqiws Simplement Feriodiques, 193
Remarque. — On peut encore pour le developpement de Un employer la
fonnule suivante,
(16)
£/» + .=
p,
VQ,
0,
0, . . . .
VQ,
p,
VQ,
0, . . . .
0,
^Q,
p,
V Q
0,
0,
■ • • ■
VQ,
p, . . . .
• ••••••••
(n colonnes),
cependant Temploi de la forniule (14) est bien preferable.
Section V.
Des relations des fonctions Un et V^ avec les fractions continues.
Les fonctions Un et F„ sont developpables en fractions continues ; en eifet,
considerons Texpression
(17)
— = Oo + —--
On Ol + «2
h + <h
h+.
(18) {
et designons par R^ et Sn le numerateur et le denominateur .de la 7i**^ reduite ;
on sait que Ton a
-^n -f- 2 = ^n — 2 -^n + 1 "T ^n + 2 -^n r
'^n + 2 = ^n + 2 *^n + 1 + ^n -f 2 '^fi r
et, de plus
(19) FnSn + l — Iin + lSn={—iy a,a^(h «* + i .
Par consequent, si Ton pose
do = bi = b2 =
ai = 02 = (h =
on obtient I'expression
(20)
..=*»= P,
• • = «n = — Qr
U."~
Q
P-Q .
49
194 Lucas, Theorie des Fonctioas Numeriqaes Simplement Periodiques.
dans laquelle n dosigne le nombre des quantites egales a P.
On a ainsi, dans la serie de Fibonacci :
(21)
1 (1 + V "))" '— (1 — V5 )'"' ^1,1
(1 + Vo)" — (1 — V5)" 1 + 1
1 + 1_
i + .
dans la serie de Fermat :
2 " ' — 1 2
(22) .:i=3_-_..
^ ^ 2" — 1 3 — 2
3"— 2
3 — .
et dans la serie de Pell,
m^ (1J-V2)- '-( 1-V2)" ^'_g_l _
^^""^ ( 1 + V^) •• - (1 - V^)- " ' 2-1
2 — 1
2 — 1
2 —
D'ailleurs, on a generalement
(24) -^i;,^' = « -
-(4)
n+ 1
?/„ 1 /ft^"
-a
done, en designant par a la plus grande des racines, prises en valeur absolue,
de Tequation (1), on a .
(25) Lim -^-'-^ = a ,
lorsque n augmente indefiniment. Cependant, nous ferons observer que ce
dernier resultat ne s'applique pas dans le cas des series de troisieme espece,
c'est-a-dire lorsque les racines de Tequation proposee (1) sont imaginaires.
Au moyen de cette derniere formule, il est facile de calculer rapidement
un terme de la serie U^ lorsque Ton ne connait que le precedent. Soit, par
exemple, dans la serie de Fibonacci
Lucas, Thdorie des Foncfions NumSriques Simflement Periodiques. 195
B„ = 7014 08733,
et _
a = ^i" '^^ = 1, 61803 39887 39894 8482 ;
si I'on calcule par les methodes abregees lo produit a. 1144, a moins d'une unitS
pres, on trouve exactement, puisque «„ est entier
u^ = 11349 03170.
On peut, d'ailleurs, determiner directement le dernier chiflfre de w„; ainsi dans
ce caa particulier, il eat facile de faire voir que deux termes, dont lea rangs
different d'un multiple quelconque de 60, sont terraines par le inSme chiffre;
si Ton suppose alors p interieur a 60, on peut demontrer que les derniera
chiffres de Up et de m, sont complementaires, lorsque la somme ^ + y est
egale a 60 ; on peut done supposer maintenant p egal a 30 ; et meme p infe-
rieur a 15, si Ton obaerve que lea termes Wu + p et u,s^^ ont les memes derniers
chiffres, lorsque p est impair, et leurs derniers chiffrea complementaires,
lorsque p est pair.
On a, plus generalement, la forinule
(26) ^iK ^ r _ ^„
dans laquelle les F). sont en nombre n, et, loraque n augmente indefiniment,
(27) Lim E<!!±11L — a' .
A la formule (26), correspond, dans la theorie des fonctions circulaires la
forraule
sin (n + 1) z -, 1
\ — '— := 2 coa z — ,
2 cos z — 1
2 cos z — 1
2 cos z — .
dans laquelle I'expression 2cosz est repetee n fois.
■Journal da Crelle, tome xvi, pag. 9i ; IB!
196 IjUCAs, Theorie des Fanctians Numeriques Simplement Periodigties.
On a aussi pour la serie des F„, la relation
Vr - (T
Vr-.
Q'
VrV
©
(30) I
(31) {
dans laquelle la quantite Vr est repetee n fois.
Les nombreuses proprietes des determinants et des fractions continues
donnent lieu a des proprietes analogues pour les fonctions U^ et F,. Ainsi
la propriete bien connue de deux reduites consecutives, renfermee dans la
formule (19) donne
V^- F„._,F„M = -Q"-^A,
et, plus generalenient
^ nr ^ {n-\)r* {n \ \)r H ^^^rl
on a, dans la theorie des fonctions circulaires, les formules analogues
sin^j- — sin (a* — y) sin {x + y) = sin^y ,
cos'^^ — cos (.r — y) cos {x + y) = sin^y ,
r
11 est (I'ailleurs facile de verifier immediatement les formules (31), en
remplayant U, V et Q en fonction de a et b. Ainsi, on a encore
AUl,r = fi"' •'' + 6** — 2Q" + ',
AUi =d^ +b^ — 2Q- ;
done, par soustraction :
et, par suite
(32) £/■„%,— Q'f/„*= UrU,n^^;
on aura, par la meme voie, la relation
(33) V!^,-Q^Vl = AUrU^^r^
La formule (32) donne plus particulierement, pour r = 1, la relation
(34) m^i — Qui= u^^x'
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Orson Pratt, (3 copies), .
•
Salt Lake City, Utah.
J. O. Wiessner,
it it
Fred Wilson, .
.
The Dallefi, Oregon.
Lucas, TMorie des Fonctions Nunieriqms Simplement Periodiques. 197
Cette derniere formule a ete appliquee par M. Gunther, a la resolution
de Tequation indeterininee
en nombres entiers * ; il est facile de voir qu'un tres-grand nombre de for-
mules de cette section et des suivantes, conduisent a des consequences
analogues, mais beaucoup plus generales.
Section VI.
I)eveloppe7)ie7it des fonctions U^ et F„ en sSries de fractions.
U V
Les formules (30) donnent lieu aux developpements de ^^^et 1^^ en
series dont les termes ont pour denominateurs le produit de deux termes con-
secutifs des series U et F. On a, en eifet.
Tin ~ u, "^ VIZ, uj "^ yu, uj •••• + + V u; u~J '
et, en reunissant les fractions contenues dans chaque parenthese,
(•^\ £^+i_^.__^ Q' _ ^_ __^!Lr.> .
yoo) ^- - -^^ ^^ ^^ ^^^^ -^^^^ .... ^^_^ ^^ ,
on a ainsi^dans la serie de Fibonacci, pour n augmentant indefiniment
1 + VJ_ \ 1 _ 1_ _1_ _ _1_ _1_ _ _1_
(60) 2 -^^1.1 1.2^2.3 3.5^5.8 8.13^
En suivant la meme voie, on obtient les formules plus generales
et
(38) Yi:^j-=^^-[.^ +^ + .... +__r_]Acr;
'"- '0 'O'r ' T ' ir '(n— Ijr'Br-*
nr
On tire encore des deux relations
(39) I
* Journal de Maihimatiquea purea et apjdiquces^ dc M. Besal, pag. 881-341 ; Octobre, 1876.
Vol. I— No. 8.— eo.
198 Lucas, Theorie de)i Fonctions Numeriques Simplement Periodiquss.
que nous demontrerons plus loin, les developpeinents
\V V "^ ^ ' L FT" ^ T V ^ "^ F F J'
^ Ml!L-r=L'_2^"i^J— L_+ _^^ I .... I ^*-'" 1.
I ^n • At t/„ t^UrUn ^. r ^n + r ^n -\- 2r ^n + (* - l)r ^n i- fcr"*
Lorsque k augmente indefiniment, les premiers membres des egalites
precedentes ont respectivement pour limites — — et V^ ; on tiendra compte
VA
dans le second membre, des conditions de convergence.
On pent ainsi developper la racine carree d'un nombre entier en series de
fractions ayant pour numerateur Tunite ; c'etait un usage familier aux savants
de la Grece et de TEgypte ; ainsi, par exemple, cette valeur approximative de
^^=l + i- + .
4 3 ^ 10 ^ '
rapportee par Columellb au chapitre V de son ouvrage de Be Susticd; ainsi
encore, cette valeur approximative de
donnee par les auteurs indiens Baudhayana et Apastamba*; cette valeur
approximative est egale au rapport des termes Fg = 577 et Us = 408, de
la serie de Pell.
Section VII.
Bes relations des fonctions U^ et F„ avec la theorie de la divisibilite.
Si nous posons a = a*^ et /? = b% et, par suite, a/? = Q^, nous obtenons,
par la formule qui donne le quotient de a" — ^'^ par a — ^, les resultats
suivants :
1°. Lorsque n design e un nombre pair :
(41) — = nn - l)r "f" V^ Mn - 3>r + V ' (n - 5)r + • • • • + W f^r J
2°. Lorsque n design e un nombre impair :
n-l
(42) ^'= F,„.,,„+ ^'F(„_,,,+ eF;,_,,,+ . ... -\-Q'-
* The QtUvasittraii by G. Thibaut, pag. 18-16. Jouri}al of the Aaiaiie Society of Bengal, 1876.
Lucas, Theorie des Fonctions Nvmerigues Simplemmt Periodiques. 199
Le quotient de a" — /?" par a + /3, loraque n deaigne uu nombre pair, donne
encore
(43) !^ = U,._„r- Q'U,._„,+ (rU,._„,- . . . . +{-(f)'-'u,,
et le quotient de a" + (3" par a + ;3, lorsque n designe un nombre impair,
donne enfin
(44) ^ = f;._,„ - er,..,,, + <rF,._.„ - . . . . + (- orf-^' .
Pour n = 2, on retrouve la formule
(3) V^r= l^rK,
et, pour w = 3, on a
Les relations precedentes nous montrent que U^ eat toujoura divisible
par ?7„, lorsque m est divisible par n; de memo F„ est toujours divisible par
V„, lorsque m est impair et divisible par n; par consequent Cf^ et F„ ne
peuvent 6tre des norabres premiers, que si m est premier ; mais la reciproque
de ce theorerae n'a pas lieu.
Dans la serie de Fibonacci, u^ est divisible par 2, Ut est divisible par 3,
Us est divisible par 6; par consequent, M3„, «,„ et %„ sent respectivement divi-
aibles par 2, 3, et 5. Ainsi encore, bien que 53 soit premier on a
Mj3 = 953 X 559 45741.
Reprenons les egalites
(6) r„+5C/", = 2a-, F, — 5P"„ = 2J";
nous obtenons, en multipliant membre a membre, la relation
(46) F^ — Afr^=4Q-,
qui correspond, en trigonometrie, a la formule
cos'z + sin'^z = 1.
Cette relation nous montre que si U„ et V„ adraettaient un diviseur com-
mun 6, ce diviseur serait un facteur de Q; mais, d'autre part,
et, en supprimant les multiples de Q, ce qui revient evidemment a remplacer
& par Q, on a la congi'uence.
(47) V„ = P" , (Mod. Q) ;
200 Lucas, Theorie des Fanctioiu Numeriques Simpleinent Periodiques.
done, tout diviseur Q de Un et V^ diviserait P et Q\ or nous avons suppose
premiers entre eux. De la resulte cette proposition :
Theoreme: Les nombres Un et V^ sont premiers entre eux.
Si Ton designe par fi I'exposant auquel appartient P suivant le module
Q, on sait que la congruence
i^ = 1, (Mod. <2),
est verifiee pour toutes les valeurs de n egales a un multiple quelconque de (/,
fi etant lui-meme un certain diviseur de I'indicateur ^{Q) de Q, ou du nombre
des entiers inferieurs et premiers a Q ; par consequent, a cause de I'egalite
(47), on resoudra la congruence
(48) F„ = 1, (Mod. Q\
par toutes les valeurs de n egales a un multiple quelconque de fi.
Section VIII.
Des forrnes lineaires et quadratiques des diviseurs de U^ et F„, qui correspondent
avj: valeurs paires et impaires de V argument n.
La formule (46) conduit encore a d'autres consequences importantes sur
la forme des diviseurs de Un et de F^, car on en deduit immediatement les pro-
positions suivantes, suivant que Ton considere n egal a un nombre ^air, ou a
un nombre impair,
Theoreme : Les termes de rang impair de la serie U^ sont des diviseurs de lu
forme quadratique x^ — Qj^.
En tenant compte des resultats bien connus de la theorie des diviseurs
des formes quadratiques, on a, en particulier, pour les formes lineaires corre-
spondantes des diviseurs premiers impairs de C/2r + i
dans la serie de Fibonacci : 4 j + 1 ;
Fermat : 8j + I, 7 ;
Pell : 4j + 1.
Ainsi, les termes de rang impair de la serie de Fibonacci ou de la serie
de Pell ne peuvent contenir comme diviseur aucun nombre premier de la
forme 4j + 3.
Theoreme : Les termes de rang pair de la sirie Vn sont des diviseurs de la
forme quadratique x^ + At/^.
(4 44
(4 4C
Lucas, Theorie des Fonctians Nunieriques Simplement Periodiques. 201
En particulier, les formes lineaires correspondantes des diviseurs pre-
miers impairs de Fjr sont
dans la serie de Fibonacci : 20j + 1, 3, 7, 9 ;
Fermat : 4y + 1 ;
Pell : 8j + 1, 3.
Th£oreme : Les tennes de rang impair de la shie V^ sont des dimseurs de la
forme quadraUqu^ a^ + Q^^^-
En particulier, les formes lineaires correspondantes des diviseurs pre-
miers impairs de Fg^ + 1 sont
dans la serie de Fibonacci : 20q + 1, 9,. 11, 19 ;
Fermat : 8y + 1, 3 ;
Pell : Sq + l, 7.
44 44
44 44
(49) {
Section IX.
I)es formules concernant F addition des fonctions num^riqties.
En multipliant membre a membre les relations
r^ + 8U^ = 2a-, r^ + SU„ = 2a\
on obtient,
si Ton change a en ft, et 5 en — 5, on deduit ensuite par addition et par
soustraction, les formules
2U^^,= U^V^+U^V„,,
2V^^^= V^V^ + AU^U,,
auxquelles correspondent en trigonometrie les formules de Taddition des
arcs :
sin (a; + y) = sin x cosy + sin y cos x ,
cos (^ + y) = cos X cosy — sin a: sin y .
Si nous changeons 7i en — n dans les formules (49), en tenant compte des
relations
(60) '^- = -|"' »'- = |''
nous obtenons
en faisant m = n -\- r, on obtient les formules (39) donnees plus haut.
(51) {
51
202 Lucas, Theorie des Fonctims Numeriques Simj)l€inient PSriodiques.
La coinparaison des egalites (49) et (51) nous donne immediatement
77 />« 77 = 77 F .
posons maintenant
il vient
(52)
m + n zz r, m — n zz 5,
r — «
2 Z
Ur—Q^U,= Ur-sVr.s]
ces relations sont entierement semblables a jcelles qui perinettent de trans-
former la somme ou la difference de deux lignes trigonometriques, en un pro-
duit. On a, de meme
(53) :
On aura encore, com me pour la somme des sinus ou des cosinus d'arcs
en progression arithmetique
(54)
m
Un+lQ*
— r — 2r ^nr - -^ r
Un. + Q'^ U^ + r + Q'^ U^ + 2r+ . . . + Q'^ U^ + nr= U^'^ +- ^
UrQ
m
-r ~2r — nr ^n -+ 1 Q
et, par suite . •
(55)
— r — 2r — nr
^m I S^ ^m + r 4" y ^m + 2r + • • • "T W ^m + nr ^m -»- ~ r
On trouve des formules beaucoup plus simples en partant des relations
'r'n + 2r = V r '^ n -\- r ^ ^n 5
si Ton remplace successivement n par 0, r, 2r, . . . (ii — 1) r, et si Ton ajoute,
on obtient
(13) {
(56)
F 4- F 4- -0- F ^r + ^ ^nr ^(n -f l)r
^ + K^ + .... + K„ _______ ,
Lucas, ThSorie des Fonctions Nunieriqiies Simplement Periodiques. 203
Ces formules se presentent sous une forme indeterminee lorsque le denomi-
nateur s'annule, c'est-a-dire pour
1 + arh" — or — b' z=0,
ou bien
(1 _ or) (1 _ ftr) := ;
c'est-a-dire pour les valours de a ou de ft egales a T unite ; dans ce cas, on
emploie le procede de sommation de la progression geometrique. On a
d'ailleurs, dans la serie de Fibonacci, pour r = 1 et r = 2,
«^i + «fj + ^ + . . . . + ^^n = ^„ + 2 — 1,
«^2 + «^4 + ^^6 + • . . . + U2n = V^n + 1—1?
Vi+ V2+ v^+ + v^ = v^^2 —S,
^2 + ^4 + ^6 +. . . . . + V2n= V2n + 1—1-
On trouvera encore plus general ement,
(F{7\ TT MTT 4-TT 4- JLTT Um^r-\-Q !^Um+nr C/„i+(n+l)r Q^ U^
\p*) ^m+r-T^m f2r-r ^m +3r "T • • "t" t/,„ ^ „,. — ~ — ,
i + y Vr
et un resultat analogue en changeant U en F.
La formule d'addition pent s'ecrire encore
2 JL^LL?* — ^ V I V -
^ n ^ n
on a, par consequent
/KQX O ^w -i- n ^m + n — 1 ' * * * ^m -\- 1 ^ ro + n — 1 ^ m+n — 2 • • • • ^m jr
^ ^ t^, u„^,: u, -£/■„ t/;_, u; '"
t ^m -t- n — 1 ^ m + »» — 2 • • • • ^w + 1 TT
On en deduit iminediatement cette proposition :
Theoreme : Le produit de n ternies conseciLtifs de la serie U^ est divisible par
le produit des n premiers termes.
Nous terminerons ce paragraphe par la demonstration de formules d'une
extreme importance ; car elles nous serviront ulterieurement comme base de
la theorie des fonctions numeriques doublement periodiques, deduites de la
consideration des fonctions symetriques des racines des equations du troisieme
et du quatrieme degre a coeflScients commensurables. Les formules (30)
nous donneilt
^m — 1 ^m + 1 = ^m ^ Q ?
204 Lucas, Theorie des Fonctions Numeriques Simplenient Periodiques.
on en deduit
et, par les formules (32)
on a, de meme
En particulier, pour m =. n + 1, et pour in =i n + 2, on a
et des formules analogues pour les F„ .
Les formules (A) et {B) appartiennent a la theorie des fonctions ellipti-
ques, et, plus specialement, aux fonctions que Jacobi a designees par les
symboles © et H.
iS)
[
Section X.
De la soiwiie des carves des fonctions numeriques U^ et V^ .
Si dans la relation suivante
V^*^) ^ ^r f 2«p ^* + 2jca ^^ '^^r -f * + 2jc (p f a) V '^^r — * + 2jc(p— a) J
nous supposons successivement x egal a 0, 1, 2, 3, . . . w, et si nous ajoutons
membre a membre les egalites obtenues, apres avoir divise respectivement par
1 Op + <^ Q2{p-V9) />(p + <r)
nous obtenons la formule
/• it = n
(60)
ETT TT ^r + 2np ^* + (2n + l)<r ^ ** ^r + (2n+ l)p ^* + 2ikr
^r + 2«p ^'a + 2«a —
je=.0
AQr''-''^U„_,U,^
^UrU.^^-Q-'Ur-^U.
On a, en particulier, pour 2p = r et 2(r = «,
AU,_,U,^,
(61)
TT TT I ^2r ^2* i ^3r ^Sa i i ^(n + l)r ^(n + 1)«
Q "
«"
« — r
^(n + l)r ^2(n + 1)# ^ ^n + 1)# ^2(n + 1),
AQ''''-"U^_^U._y,
(62)
Lucas, Th^orie des Fonctions NumSriqaes Simjplement PSriodigues. 205
et, plus particuliereinent encore
tv , t/2. , C/2. , , ^n + l)r _ 1 r C^2n + S)r 9 q"l
^^ I _^ 4. ^ J. J. ^(2» + i) >- — ^ r ^4(n + l)r O^ ol
Par un precede analogue, on trouvera aussi les valeurs de
K » JC »
En particulier
(63)
Zl -L J^ -L Xk 4- 4- ^'* — 2ii 14- ^(^ + ^>-
F2 F2 Tr2 Tr2 rr
^r I _18r , r5r , , ^(2n4-l)r 2w 4- 2 4- ^^<»->-^)^
/y ~ /}3r I /)5r ' ' • • I /)(2« + l)r — ^l^ T ^ T jj p.^ ^ j)
On a aussi, dans le cas general,
/^ jc°-n
(64)-
ETT2 ^2m + 2(»-hl)r ^2»> ^ L^2m4-2i»r ^2m— 2rJ __^ O/lm
"'■•'" A(F^— (^* — 1) ^
ETr2 ^2m + 2(n + l)r ^Tm Q> \_ ^2m + 2inr ^2m— 2rJ 1
„ "■ '"" F^-Q^-1 +
(Yn + iyr 1
On a, par exemple, dans la serie de Fibonacci
«?-< + t4-...-(-l)-'t*i = y[2n + l-(-l)-'^^],
«^ + wi + «^ + . . . + «(l. + i)r= i [^^^^!^i^' - (— 1)"- (2n + 2)] ,
vi + vl + vl^ . . . -V v,l^,,r = '^:^^-±^ - {^
Vrtr
la formule plus simple
(66) t^ + «4 + ^+ . • . + «^n = «*nWn + i,
donne ainsi, pour le c6te du decagone regulier etoile, cette expression
(65)
(67)
p p + r ' r+r+2^ v+v + ^ + 2^
1
+
52
1* + 1^ + 2* + 3* + 5^
206 Lucas, Theorie des Sanctions Numiriques Simjplement PSHodiques.
On a encore, dans cette serie
,ggx I ^l^l + '^h^h + ^^4 + + 'Ihn-iU^n = Ul, ,
WiW^ + Wi«^ + ^-3^4 + + W2nW2» + l= Win + l — 1.
Section XI.
Des relations des fonctiom U^ et Vn avec la theorie du plus grand cornmun diviseur.
Nous avons trouve la formule
par consequent, si un nombre impair quelconque 6 divise U^^^^ et U^j il
divise U^ V^ ; mais nous avons demontre (§ 17) que TJ^ et V^ sent premiers
entre eux ; done divise TJ^. Inversement, tout nombre impair qui divise
IJ^ et TJ^ divise ^„, + „ ; done, en ne tenant pas compte du facteur 2, on a cette
proposition fondamentale :
Theorem E : Le plus grand commiin diviseur de U^et de U^ est egal a Ud^
en designant par D le plus grand cornmun diviseur de m et de n.
En particulier, les termes U^ et U^ sent premiers entre eux lorsque m et
n sent premiers entre eux, car JJi est egal a I'unite. On deduit d'ailleurs du
theoreme fundamental un grand nombre de propositions entierement sem-
blables a celles que Ton obtient dans la theorie du plus grand commun
diviseur et du plus petit multiple commun de plusieurs nombr^s donnes.
II resulte encore de ce qui precede que, dans la recherche du plus grand
commun diviseur de deux termes 11^ et ?7„, les restes successifs ferment aussi
des termes de la serie ; en particulier, les restes successifs de deux termes
consecutifs donnent, dans le cas de Q negatif, tons les termes de la serie
decroissante, a partir du plus petit d'entre eux. Lam£ * a observe que, dans
la recherche du plus grand commun diviseur de deux nombres quelconques,
le nombre des restes est au plus egal au nombre des termes de la serie de
Fibonacci, inferieurs au plus petit des deux nombres donnes, et il en a deduit
ce theoreme :
Le nombre des divisions a effectuer daiis la recherche du plus grand commun
diviseur de deux nombres donnes est au phis egal^ dans le systhne ordinaire de nume-
ration^ a cinq fois le nonibre des chiffres du phis petit des deux nombres donnes.
* Comptcs renduB de I'AcadSmie des Sciences de Paris, t. zix, pag. 868. Paris, 1844.
Lucas, TMorie des FoncUons Numdri^ues Simplemmt PSriodiques. 207
On trouverait une limite plus rapprochee, en calculant par logarithmee
le rang du terme de la s^rie de Fibonacci immediatement inferieur an plus
petit des nombres donnas. On voit aisement qu'il suffirait, en designant ce
plus petit nombre par A, de prendre le plus petit entier contenu dans la
fraction
log ^ + log V 5 _ log A + 0.349
, 1 + V5 ~ 0.209
log -jr__
Mais il est preferable de s'en tenir a la limite donnee par I'elegant theoreme
que nous venons de rappeler.
Section XII.
De la multvplication deafonctUma numiriguea.
On peut esprimer les valeurs de U^ et V„ qui correspondent a toutes les
valours entieres ot positives de n, en fonction des valeurs initiales ; en effet,
'on a success! vem en t, pour U, par ezemple,
U,= {P'-Q)U,— QPU„,
U,= {P^ — 2PQ) U,~Q {P' —Q)U,,
U, = {P* — SP'Q +^)U, — Q {P' — 2PQ) U, ,
On obaervera d'abord que si (p, designe le coefficient de Ui dans f/„ + 1 , on a
en general,
Le coefficient ^„ est une fonction bomogene et de degre m, de P et de Q, en y
considerant P au premier degre et Q au second. Si Ton forme le tableau des
coefficients de ^„, on retrouve aisement le triangle arithmetique, raais dans
une disposition speciale. On a d'ailleurs, ainsi qu'on peut le verifier a
poateriori
'™' 1 _ («-8)(»-4)(n-S)
1.2.3 ^'^
208 Lucas, Theorie des Fonctions Nunieriques Simpleinent PModiques,
(71) I
et, en raeme temps
£^i»+ 1 = <?>n ^1 Q^n-\ Uq ,
Fn + i =<?>nFi— $<^^_iFo,
avec les conditions initiales
J7o = 0, U, = l, Fo = 2, V,z=P;
par consequent, on a encore
<?>n = ^n + 1 •
On a, en particulier, dans la serie de Fibonacci, pour PzzletQzz — 1,
(72) 1 + C^-.i^i + <7»-2,2 + ^n-8,3 + = ^i»>
et, pour P = 1, Q = 1 ,
(73) 1 ^n-1,1 + ^n-2,2 ^n-8,8 + . • . = — =r SlU — - .
V o ^
Les formules precedentes se generalisent aisement par la consideration
de I'equation (8). En effet, si Ton pose
(74) ^„= F,~ — "^li V;-'Q\+ {n — 2){n — 3) yn^^Q^
1 1 • ^
_ (n — 3)(n — 4)(w — 5) y,_,^ ,
1.2.3 ' V"-h...
on obtient, comme ci-dessus,
Un + Hnr^'^n-l^m + r (t'^»-2^mi
et^ pour m ^ 0, on a encore la relation
(76) ^-' = ^
qui permet de calculer inversement la fonction 4' a I'aide des valeurs de U.
D'ailleurs, cette relation, dans laquelle n designe un nombre entier, a lieu
quelle que soit la valeur de r ; on a ainsi, pour r = 0, la formule
(77) n = 2»-^ — a.-2,i2~-^ + C;.-3,22»-^ - 0.-4.32""' + • . . .
Nous ferons observer que les resultats precedents correspondent aux deve-
sin 7iz
loppements bien connus de —. et de cos 7iz suivant les puissances de cos «,
obtenus pour la premiere fois par Viete.*
(76) {
« Opera, Lejde, 1646, pag. 295-299.
LucAg, Theorie des Fmictions NumeriqU'es Simj)h)nent PSriodiques. 209
Section XIII.
De la loi de la rSpetition des nombres premiers dans les series r^currentes
simplement periodiques.
Nous exprimerons encore les fonctions U^p et Vnp, en fonctions entieres
de Un et de Vny par des formules analogues a celles qui ont ete donnees par
MoivRE et par Lagrange *
En eflfet, si Ton designe par CZ lo nonibre des combinaisons de m objets
pris w a n, on a la relation suivante :
(78) a^ + /3Pzz(a + i3)^~|-a/3(a + /3)^-^+|- ^;.3a^/3Ma + /?)''--'+ • • • •
r '
que Ton pent verifier a posteriori^^ et dans laquelle tous les coefficients sont
entiers, puisque Ton a
— ^p — r — 1 ^j,_r — 1 "t" ^p — r — 1 •
Posons, dans I'hypothese de p impair^
a = a'' ei(3 = — b\
nous obtenons
+ ^ C;z\.,Q''^-''-'U^n-'' +
r
La formule precedente conduit a la loi de la repetition des nombres
premiers dans les series recurrentes que nous considerons ici. Dans la serie
naturelle des nombres entiers, un nombre premier p apparait pour la pre-
miere fois, a son rang, et a la premiere puissance ; il arrive a la seconde
puissance au rang ^^ a la troisieme au rang p^, et ainsi de suite ; de plus, tous
les termes divisibles par ^" occupent un rang egal a un multiple quelconque
dej?*. Mais dans les series recurrentes simplement periodiques, il n'en est
pas completement ainsi. Nous demontrons plus loin que les termes de
celles-ci contiennent, a des rangs determines, tous les nombres premiers;
* Commentarii Acad. Petrop., t. XIII, ad annum MDCCXLI-XLIII, pag. 29. Le9on8 sur le calcul des
fonctions, pag. 119.
53
210 Lucas, Theorie des Fonctions Numeriques Simplenient Pinodiques.
raais si ces nombres premiers p n'apparaissent pas, pour la premiere fois, dans
la serie au rang p, cependant ils s'y reproduisent a intervalles egaux a ^,
comme dans la serie ordinaire, et I'apparition de leurs puissances successives
se fait comme dans la serie naturelle. Ainsi, en general, dans I'etude arith-
metique des series, deux lois sont a considerer : la loi de T apparition des nom-
bres premiers, et la loi de la repetition.
Nous demontrerons, pour I'instant, que la loi de la repetition est identi-
quement la memo dans la serie naturelle, et dans les series des ?7„. En effet,
si p designe un nombre premier, et ?7„ le premier terme de la serie divisible
par p^^ on observera que le dernier terme de la formule precedente est divisi-
ble par p^ ^ S et non par une puissance superieure de p\ on a done la proposi-
tion fondamentale suivante :
Theoreme : Si X designe le plus grand exposant d^un nombre premier p con-
tenu dans U^ , V exposant de la plus haute puissance de p^ qui divise Uj^, est egal a
X + 1.
Ainsi, par exemple, dans la serie de Fibonacci, u^ est divisible par 7 ;
done W56 est divisible par V et non par 7^ ; dans la serie de Pell, Uq ei v^ sont
respectivement divisibles par 13^ et par 31^ ; done U^i et U^ sont divisibles
par 13^ et par 31^ et non par des puissances superieures.
Inversement, si a^ i b^ est divisible par p\ a ± 6 est divisible par |?^"^ ;
ce resultdt donne des consequences importantes dans la theorie de Tequation
indeterminee
^ + y^ + ^ = ,
dont rirresolubilite, non demontree jusqu'a present, constitue la derniere
proposition de Fermat.
Section XIV.
Nauvelles formes lineaires et quadratiques des diviseurs de U^ et de F„ .
La formule (79). donne, successivement, pour p egal a 3, 5, 7, 9, . . . les
formules suivantes
U^ = A'U:+ 5QrAU„'+ 5Q'^U^,
U,^ = A'U: + 7QrA'U,'+ UQ'^A Cr?+ TQ^^'U,.
(80)^ U,n = A'U:+ 9QrA'U:+27Q'''A'U:+30(tAU^+ 9Q^U^,
Uun = A^ U],' + nor A' U: + 44Q2"A^ U: + 77Q^A2 U^ + 55Q^"A U^
V.
Lucas, Theorie des Fonctions NumiriqueB Simplement PSriodiques. 211
On a ainsi
(81) ^i2 = AC/2 + 3e»,
et, par suite, la proposition suivante :
Th^oreme : Les diviseurs de -~ sont des diviseurs ds la forme guadraUgue
Ax' + SQrf.
En particulier, les formes lineaires des diviseurs premiers impairs de ~ sont
pour la serie de Fibonacci : SOq + 1, 17, 19, 23 ;
de Febmat : 6^ + 1 ;
de Pell : 24j + 1, 5, 7, 11 ;
et les formes lineaires des diviseurs premiers impairs de ^.JpLll) sont
pour la serie de Fibonacci : 60j + 1, 7, 11, 17, 43, 49, 53, 59 ;
de Febmat : 24j + 1, 5, 7, 11 ;
de Pell : 24^ + 1, 5, 19, 2'3 .
On a aussi
(82) . 4j^ = i2AU; + 5Qy — 5(^,
et, par suite :
U .
THisoB^ME : Les diviseurs de yp sont des diviseurs de la forme quadratique
Les formes lineaires des diviseurs premiers impairs sont, dans les trois
series prises pour exemples,
20q + 1, 9, 11, 19.
Nous avons aussi
(83) 4^^ = A12AUI + 7QrU^y + 70^7,%
et, par suite :
Th^obeme : Les diviseurs de -y^ sont des diviseurs de la forme quadratique
Supposons maintenant que p designe un nombre pavr^ et faisons encore,
dans la formule (78),
a = a", /3 = — 6^
212 Lucas, Theorie des Fonctions Numeriques Simplenient Periodigties.
nous obtenons
(84) V„, = h^m + |- Q'h^^'Ur' + |- C^.^Q'^Sp-'m-' + ....
+ ^ c;zj«iQ~-5^-^t/5-^ +
r
On a, en particulier, pour p =z 2, la formule
(86) V,^ = £^Ul+2Qr,
et, par consequent, la proposition suivante :
Theoreme : Les diviseurs de V^n smt des diviseurs de la fomie qiiadratiqu^
Ax" + 2QY .
Les formes lineaires correspondantes des diviseurs premiers impairs sont,
pour n pair
dans la serie de Fibonacci : 40? + 1, 7, 9, 11, 13, 19, 23, 37 ;
de Febmat : 8j + 1, 3 ;
de Pell : 4 j + 1 ;
et, pour n impair
dans la serie de Fibonacci: 40? + 1, 3, 9, 13, 27, 31, 37, 39;
de Fermat : 4j + 1 ;
de Pell : 4j + 1 .
On devra, dans les applications, combiner ces resultats aA^ec ceux que nous
avons donnes dans la Section VIIL
Faisons enfin dans la formule (79),
a = a% /? = 6%
nous obtenons, en supposant indifferemment que p est egal a un nombre pair
ou a un nombre impair :
(86) r„, = Fs - -|. orvr' + -f ^j-s^^Fr^ - . . .
On a ainsi, en faisant successivement p egal a 2, 3, 4, 5, 6, ... les resultats
suivants
(87)
Fe„ = V: - 6QrV: + 9Q!^V„' - 2Qf^ ,
qui conduisent encore a des formules entierement semblables aux precedentes.
Lucas, Tkeorie des Fonetions NumMques Shnpleinent Periodiques. 213
Section XV.
Bes relations des fmctions Un et V^ avec ks radicaua: continus.
On tire de Tequation
x' = Px^Q,
la formule
et, successivement
x = ^^—q + P V — "QinPo:
^ = y\—Q + P^—Q + Psf — (i + Pjc
» >
par consequent, puisque Ton peut supposer P positif, on a, pour Q negatif,
(88) a = Um.yl_ q + P V" ^ + ^ ^— ^ + • • • • »
a designant la racine positive de I'equation proposee. Ainsi. dans la serie
de Fibonacci
dans la serie de Pkll,
1 + V2 = Lim.^l ^2 a/i + 2V 1 + 2^^1 + .... ,
et, dans la serie de Fermat,
2 = Lim. ^2 + A/2 + V2 + V2 + ..r^ .
On salt que ce dernier radical se presente dans le calcul de 7t, par la methode
des perimetres, iinaginee par Archimede.
Mais les resultats obtenus dans la section precedente, conduisent a des
formules plus importantes, qui trouveront leur eniploi dans la recherche des
grands nombres premiers. On tire, par exemple, de la premiere des for-
mules (87)
F„ = V2^" + r,:;
et, de meme, en changeant n en 2n, 47i, 8n, . . .
V^ = V2^ + r,«„ ;
54
214 Lucas, Theorie des Fonctiom NumMques Simplement Periodiqms.
et, par suite,
/•
(89)
V. -
V2«- + V2(^^ + \'2QM^^^T^,
et ainsi indefiniinent. Oes formules sont analogues a celles que I'on obtient
n
71
n
71
7t
pour cos — , cos -g , cos — , cos — , . . cos — .
La seconde des relations (87) donne de la meme fa§on
(90)
V = \'>.(y* -i- ^';
2Q" + VSQ*" + '^- ,
2il
n
= \2Q- + \i
*^
2Qr + \2^ + \ZCf' + ^" ,
ces formules sont semblables a celles que Ton obtient pour cos — , cos — ,
cos|^,...cos^
La troisieme des relations (87) conduit encore d des 'formules qui corres-
pondent a celles qui donnent cos — , cos ^a' ^^* :?7S' * • ^^^ k or» ^* 9\xi^\ de
quelques autres.
10' ^ "'2()'™40' b,2'
Section XVI.
Devehppements des puissances de Un et de F„ en fonctions lin^aires des fermes dont
les arguments sont des multiples de n.
On pent exprimer les puissances de Z7„ et de F„ en fonctions lineaires des
termes dont les rangs sont des multiples de n, par des formules analogues a
celles qui donnent les puissances de sin z et de cos z, developpees suivant les
sinus et les cosinus des multiples de Tare z. En designant d'abord par ^ un
nombre impair^ le developpement de (a — /?)'', donne
Lucas, Theorie des Fonctions Numeriques Simplement Phiodiques, 216
et, par suite, en faisant
on obtient la formule
a-zz a;
^ = b\
(91) A^Ujl=U,
y«
p{p—l)...
P + S
^-«^v,.
^•^•"^ i.2....t:
2
On a successivement, pour p egal a 3, 5, 7, 9, . . .
A'U: = U,„ — 7QrU^ + 21Q''U^ - SBQ^'U^,
A*U:= U^ — 9Q-'U,„ + 36(tU^-S^(rU^ + 126Qr'U,,
Le developpement de (a — ^Y donne encore, en supposant maintenant que
p designe un nomhre pair:
(92)
(93)
AV5=r^--f^F;,-,,+^^f^)
Q^F(p_4),
12 3 '^cp-e)!* T^ • • • T^ • • • it y ,
12^
et, pour p successivement egal a 2, 4, 6, 8, . . .
(94) \ A'U:=V,^-6Q^V,^ + 15Q^V^-20Q^.
A'Ul= V^-8Q^V,^ + 28«^F,„- 56Q^ F^^ + 70Q^,
Le developpement de (a + /?)^ donne, dans I'hypothese de p egal a un
nombre impair^
(95) F5 = F^ + |- Q» F(,_.), + ^^^~^^ «^ >^,-4)n
r ^^ P + ^
,pip — l ){p—2) ^y
.^';::in
1 • ^ . • .•
p
Q " -F,,
216 Lucas, The&rie des Fmictions Numerigues Simplement Periodiques.
et, plus particulierement
F,'= V^ + bQ'V^ + lO^V^,
(96) \ V,J = F,. + 7QrV^ + 21(rr^ + 35^V, ,
F,» = V^ + 9Q"F„ + 36Q^F«„ + M^V^ + 126^^,
(9
De meme, lorsque p designe un nombre pair,
' n = Fp„ H 7" ^" '(p — 2)n ~r — ^ 5 ^ ' (P — 4)
1.2.... (|-)
on a, plus particulierement,
^ F?=n„ + 2Q-,
F?=F,, + 4(2-F^ + 6(2^-,
(98) ^ F? = Fe. + 6Q: F,.. + 15^'- F,, + 20Q»- ,
F„« = F8„ + 8(2- Fe, + 28(?"F,, + 56Q*'F,, + 70(2*" ,
Les relations (91), (93), (95) et (97) sont elles memes des cas particuliers des
formules suivantes :
n(w — l)(n — 2)^,,^.
' r '^(m-n)r ^mr T^ "T" '^ ^(m — 2)r T :i ^ ^ r (» _ 4) r
(99)
w(n — l) (n — 2) ^^
"^ 1 9Q MU'(in— 6)r l" • • • •
An iT2n rr — TT ^^^^ (Y TT 4. 2/1 (2;i 1) ^jr 77
^-* ^r ^(m-2n)r ^ mr 1"^ ^ ^(m — 2)rT ^j ^ W, ^{m — 4,)r ....
A"772»F — F ^^ O^F -4- 2^ (2^ — 1) /)2r T^
An rr2n+l JT JT 2n'\-\ ^jj (^^ "1" !)• 2W x^2r rr
^-* *^r C'(wi— 2n— l)r ^mr ^ '^ ^(m — 2r) "T 1 ""o ^ ^(m— 4)r •••
A«/72«+iF — F —^'^"'"^O^F , (2rt + l)-2n ^g ,;.
^-^ '^r ^(m— 2i»— l)r ^ i»r ij — V ^(m— 2r)T^ ^ ^^ H '^(;pi-4)r •••
Lucas, ThSorie des Fonctions Numerigues Simplement PModiques. 217
Ces relations trouvent principalement leur emploi dans la sommatioa
des puissances semblables des fonctions U^ et V^ . Le developpement de la
puissance d^un binome donne encore lieu a un certain nombre d'autres.
Ainsi, on a, par exemple
a = a + /3 — /? et (3 = J+~a — a ;
done, pour p 6gal a un nombre impair
on a ainsi, en ajoutant et en retranchant, apres avoir pose a = a" , j3 = J* ,
les forniules suivantes :
(100)
9F — FFp P V Vp-^ j? ( j? 1) y T7JP-2 I -L ^ F V F
c\ — P TT vp-i_1 P \P ^) 7/ F^-2 I ^ rr F
On trouvera des developpements analogues pour y egal a un nombre pair, et
d'autres encore a Taide des identites
a = a — /? + /3et/3 = /3 — a + a.
La formule suivante, que Ton pent deduire du ProhUme des partis
= a^ [(a + py-' + -f (a + ^Y^'^ +^-^^±^ (a+/?)'-»i^+ . . . + C^TJ-^/?'-^]
donne en changeant a en /?, puis par addition et par soustraction,
27P+,-i_ v,^vi-' + 1- (2-F;,_„„Fr^ +^J|+i) (^F;;_,)„rr'
i — a " ^(,q—i)n'» ••• ^p + q-v4 ^{q—p + \)n'
55
(101)
218 Lucas, ThSorie des FoncUons Numiriques Simplement Periodiques.
On obtiendrait deux autres formules semblables aux pr^cedentes, en posant
a = a" et /3 = 6** ; on simplifie ces formules, en faisant jp = j.
Section XVII.
Autres formules concernant U dheloj^ement des foncUons numiriques U^ et F„ .
Considerons les fonctions a et /? de z ,
« = (,- 2 ;, /3=( );
on tire, en differentiant
doL n
adz V 2;^ — 4A '
et, en faisant disparaitre le radical
Une nouvelle diflFerentiation nous donne
^ ^ dz^^ dz
il est d'ailleurs facile de voir que les fonctions /?, a + /^ ®t a — (3 verifient la
meine equation diflferentielle. On a done, en designant par / (z) Tune quel-
conque d'entre elles, par I'application du theoreme de Leibniz
et, pour 2 = 0,
Si Ton suppose z =z F^ , A = Q^ , la formule de Maclaurin nous donne, pour
npaivy les deux developpements
(102)
__■, n" T7 , n'{n^—2') V* _ n\n^—^){n'~
^,;~ 1.2 2*(?"^ 1.2.3.4 2«Q* 1.2.3.4.5.
6
U„ _nV^_ n(n'— 2^) J?_ , «_K— 2*)i«l— 4*) J^ _
- "120^ 1.2.3 ^q^^ 1.2.3.4:6 ^Ct
+■•,
et, pour n impair
u^ _ ^r fi _ «^-i' v^ . (n'-i')K-y) f;« _ 1
!LpL— ''••.L 1.2 2*(r 1.2.3.4 2*<r "J'
K, _ T. \^_ n{n'-V) Vr' , n(n'-r)(«'-y) F;^ _ "I
~r^~'^'L 1.2.3 2*0^^ 1.2.3.4.5 2^0^ "J
(103)
Lucas, ThSoHe des FoncUons NumSriques Sinvplement Periodiques. 219
On peut d'ailleurs verifier ces formules, et les suivantes, a posteriori, en
observant que si Ton pose
G^^,, = (m^ — 2'){m' — 4*) . . . (m^ — 4x^) ,
jy^,.= (m^-r)(m«-30 ... (m^-(2x-l)«),
on a les relations
ma^^,= {m — 2x) 5'^ + i,,z= {m + 2x) ir^_i,«.
Au lieu de developper les fonctions U^r et F„^ suivant les puissances de
Vrj on peut aussi les developper suivant les puissances de Ur] on trouve
ainsi, pour n pair
(104)
2qT~ "^1.2 2^0'"^ 1.2.3.4 2*0^ "^ 1.2.3.4.5 2«Q* "^ "'
?7".. _ g7^r , n(n'-2») act; n(7t'— 2')(n^— 4^)A'Z7,V l
(;_i)r- 2 L "^ 1.2.3 2*0^"^ 1.2.3.4.5 2*^*' "^ ' "J '
et, pour n impair
V.
(105)
»* — PACT? (n'
2
r){n^ _ 32) A* Z7,*
2 2^Q'
(w*
+
1.2.3.4 2*Q''
p)(n'' — 3')(n' — 5') A" t/"/
1.2.3.4.5.6 2*0*
+ ..],
Q
^ - 77 r 4- ^ (^' — 1') A^' 4. n (n" — 1') (n^ — 3') A^U*
_ ,|_wi- ^^^ 2'Qr 1.2.3.4.5 2*(t
+
n (n* — 1^) (n* — 3*) («* — 5*) A" U^
6
1.2.3.4.5.6.7
En ayant egard a I'une ou I'autre des relations
ri=r„^ + 2Qr, et aui = v^r — 2Qr%
on obtiendra de nouvelles formules, et ainsi, par exemple :
3.4.5.6
2«a^
+ ...].
(106)
r) AUl , ri'{n^ — V){n ' — 2*) A^ U*
Q(«-2,r^2 — " . 3,4 or ' 3.4.5.6 2'Q.
On peut d'ailleurs mettre cette derniere formule et quelques autres sous
une forme assez remarquable, en observant que Ton a, pour m quelconque et
n entier positif, I'identite
m'(m'— P)(m''— 3')...(m'— (n— 1)') _ {m-^) (m—n+l) {m-^n+2) . . . (m+n—l)
3.4.5 2n ~ 1.2.3 (2n)
, (m — n + l)(m — n + 2) . . . (m + n)
1.2.3 (2n) ■
220 Lucas, TMorie des Fonctions Numeriques Simplement Periodigues.
Par consequent, les coeflScients de la formule (106) sont entiers, et Ton a
n^ K - 1^)(^^ - 2^) ... (^^ -T=rv) _ ^
3 4 5 (2r) -On^..x + o„ + ,.
Nous ferons observer que les formules (104) et (105) subsistent encore
pour des valeurs quelconques de n; on a alors des developpements en
series convergentes, lorsque ^~ n'est pas superieur a I'unite; en effet,
si Ton pose
le rapport d'un terme au precedent finit par devenir negatif (pour A positif ),
et inferieur a I'unite en valeur absolue. Cette condition est remplie pour
r = 1 dans la serie de Pell ; on a done, quelle que soit la valeur de n
r I [K.-+ ir + (v^- 1)"] = 1 + j^ + 1^^)
(107)
4
_^ n^ (w» — 2^) (w" — 4^) ^
"^ 1.2.3.4.5.6 ■^* ■'
|[,V.+l).+(V.--l,]=„+^.^)+«J^=^(^) + ....
Section XVIII.
DSvehppements en sSries des irrationnelles et de leurs hgarithmes nepSriens.
Les developpements des fonctions en series, par la formule de MacLaurin,
donnent lieu a un tres-grand nombre de formules nouvelles, pour le developpe-
ment des fonctions numeriques que nous considerons ici, et par suite, pour
celui des fonctions circulaires et hyperboliques. Lorsque les series correspon-
dantes ne sont convergentes que pour les valeurs de la variable dont le module
est inferieur a une limite donnee, on pent toujours supposer que cette variable
X est choisie de telle sorte que la serie represente la fonction, pour toutes les
valeurs de x dont le module est inferieur a I'unite. Soit done la serie
F {X) = ^0 + ^1^ + ^2^ + ^3^ + ^4^' + ;
* £n d^ignant par a le r6sidu de n' suivant le module p^ premier avec n, on d6duit de cette identity une
demonstration imm^iate d'une proposition contenue au No. 128 des DUquisitionei Arithmetical,
Lucas, Theorie des FoncUons Numeriques Sinijplement Periodiqu^s. 221
on aura, en supposant z positif,
+ 4
(1 + zf '^'''{1 + zf
1
+ •.,
et, par consequent :
+
z
+ 2
(1 + ^)
(1 + Z)
VI + J ^ \n- J —
^,1^' + ^. '
^U^3^
.3
Si I'on designe par a la plus grande des racines, supposee positive, de Tequa-
tion fondamentale (1), par r un nombre^air, ou un nombre entier quelconque,
suivant que la racine b est negative ou positive, et si Ton pose z =. — , on
obtient
(108)
^ (^) + -^ Of»') = 2^" + -^^ t; + ^' T? + ^' -^: +
Si I'on suppose z •= , on obtient deux developpements analogues aux
precedents ; ces developpements sont parfois, tres-lentement convergents ;
mais leur etude conduit a des proprietes importantes dans la theorie des
nombres premiers.
Le developpement du binome (1 — x^ donne ainsi, pour m quelconque,
les series
m{m — l){m — 2) V^
(109)
( Vn^ _ V — -Zr -^ m {m — 1) T^, _
Vr ' 1 Vr 1.2 Vr'
JJ^r ni Ur m{m — 1) U2r X VI {m — 1) (^ — 2) Til
1.2.3 |/;3+- •'
vr iVr 1.2 f; ■ 1.2.3 f;
que Ton aurait pu deduire de la serie de Bernoulli ; pour m := — 1 , on a
(110)
v.
56
Tr2 V V V
~Q'~ ^0 -h ^^ -+- ^, + ^ + . . . . ,
f/ 2r 2_r -I- ^^ I ^;_3r i ^^ i
'W~V, Vr''^ Vr' "^ V* "^ ' *
• •
222 Lucas, Theorie des Fonctions Numeriques Simplement Periodiques.
et, par exemple, dans la serie de Fibonacci
(111)
^-^ + •3 + 9" + 27 + 8i + --"
3^9^ 27 ^81^ 243^"'
les numerateurs de ces deux series de fractions sont donnes par la relation de
recurrence
On obtiendra des formules semblables pour m = ± — ; le developpement de
(1 + ^)- ± (I — x)^
donne des formules analogues aux relations (109).
Le developpement de Log (1 — x) donne les formules
/■
(112)
Log Q. - 1 + 2 1^. + 3 ^3 + 4 IT. + • • •
1 U.
k
Log ^. -2VA |__ + __ +_ _ +__+.. .J ,
1 X
celui de Log — - — donne
1 + ^
(113) Log| = 2VA[| + ^5^^ + -^
J. A^.
3 ■ F,*
et, dans la serie de Pell
(114)
La formule
It 2+k_,r 1
V2-Log(l + V2)=l + ^- + ^- +^1-^ + ^1-
l" • • • •
+
2.4
2.4.6 h*
h* 3 (z*— Ay ' 3.5(2*—**)" 3.5.7 (z*— A*)
dans laquelle on suppose
, + ..],
2.4.6 A»f7«
ni5^ Tng^'^-^^^-ri — 2 ^Ur'^2.4A-'Ur'_ 2.4.e i
donne encore
24Q4, + • • J ;
pour que cette serie soit convergente, on doit avoir A 11* ^ 4tQ^ ; on trouve
ainsi, a la limite de convergence
(116) Log(l + V2)=V2ri-- + ^-i-^-^S?-:i:^®- 1
(iio) j^og^i-t-v^; v^L^ 3^3.5 3.5.7^3.6.7.9 •J'
Lucas, Tkeorie des FonctUms NumMques Simplement Periodigues. 223
Les developpements de arc sin z et de (arc sin zy donnent, de meme,
1 AU,^ . (1.3)» A^u;
(117)
.2r
Log^ =
a" VAUr r
4 ^^ Qr~ 2'Q'
1^
2
A t^* ,
1.2.3 2^Qr "^1.2.3.4.5 2*Q-^
2_ A^ U* , }^ 2_A A'Ur'
3 2*Q^'"^ 3 '3.5 2f'(t
• • • • I 9
et, a la limite de convergence,
(118)
Log (1 + V 2) = 1
Log* (1 + V~2) = 1
1.2.3
+
(1.3)
2
(i.3.5y
2 ' 3 "^ 3 "3.5
1.2.3.4.5 1.2.3.4.5.6.7
1 2.4.6
+ ..,
+ ...
5 3.5.7
La formule remarquable de M. Scholtz conduit au developpement
1\A'U*
2'(t
/iim T ^<^ A*f7;n 3-3/i , l\AUr^ , 3.5.3/. , 1 , 1\
(119) Log3-= __^Ll___(l + _)_ + ___(l + _+_)
3.5.7...(2n— 1)3 /^ ^ . ^ ■ 1 \ A"-' U^'-^ 1
4.6.8 . . . 2» (2»i + 1) V "^ 3* "^ 5*"^ ■ ■ ■ (2«— 1)V 2«-^Q("-»)'- "*" " * J '
3.3
(1+1) +
3.6.3
(l + i +
5V
• • • •
et, a la limite de convergence
(120) L«g-(1 + V^ = l-^^,. ,3., ,^^,,-,3.
Si I'on developpe, par la formule de Lagrange, I'une des racines a'' ou b'
de I'equation
z^ — zVr+(t = 0,
on trouve
., _^ , e^ , 4 (T , 5.6(r ,
""K'^l?"^2l7"^2:3^"^ '
3 ~ 2f; "^ r* "^ 2 F/ "^ 2.3 r," "^
(121) ^
»— »•
Si Ton fait encore, par la formule de Lagrange, le developpement de y
suivant les puissances de z, en designant par y Tune des racines de I'equation
.=2+f.
224 Lucas, Theorie des Fonctions Numeriques Simplement Periodiques.
on obtient
/ 2 \" _ i _ ^ _i , n (w + 3) / z y w (n + 4) (n + 6) (z\^
\l + Vr+z/ ~ 14"*" 1.2 V4/ 1.2.3 V4/
+
n (n + 5) (n + 6) (n + 7)
vt) ~ • • • '
1.2.3.4 V4
et, en posant
4 ~ n*'
on a
nooN ^''.^«"' - 1 _i_ »» ^' _i_ ^("4- 3) CjT- , n (« + 4) (n + 5) Q^
^ ^ ~W ~ ^TTy 1.2 T^"^ 17273 17 + ---'
cette serie est convergente pour -^^ <; 1 ; elle contient la generalisation de la
^ r
formule (84).
On a encore
F,-VF;-4^
2 '
et, en developpant le radical par la formule du bin6me,
(123) 6 _ _ _ + _ __ + 2^4^ --^ + . . . ,
puis, a la limite de convergence,
(124) V2--l=^-g»- + ^Jji^-gl^^ + ...
En appliquant la formule de Burmann au developpement de z suivant les
2z
puissances de , on obtiendrait, pour tout module de z inferieur a Tunite,
L •\- Z
If
et faisant ensuite z =. — , la formule (121) donuee ci-dessus.
a*"
Section XIX.
Sur le calcul rapide des fractions continues periodiques.
On perfection ne, d'une maniere notable, le calcul des reduites des frac-
tions continues periodiques au moyen des formules suivantes. M. Catalan
a donne les relations :
LuCASy Theorie des Fonctions Nuraeriquea Simplement PModiques. 225
z z* z -^ z^ + z^
— -i + 4^ ri =
z z^ z' _ z + z' + z^ + z' + z^ + z^ + z^
1 — 2«"^1 — Z*"^l— i5«"" 1 — Z« '
z z^ z^ 4-_il_ _ g + ^' + + z'^ + z''
z'
on a, plus generalement.
2 f/^"^ 1 « ^2*
Z Z^ . , g 1 Z 5J'
■« + :: . + • • • +
2"
1 — z' 1 — Z^" l — zl — Z
par consequent, si I'on fait z = — , on obtient la formule
a""
Lorsque n augmente indefiniment, on a, pour les series de premiere et de
seconde espece,
(126) ^=|. + _^ + _|+
Par exemple, dans la serie de Fibonacci, pour r = 1,
(127) 1-V^ __ 1.1,1. 1 . 1 I .
^ ^ 2 ~ 1^3 ^3.7^3.7.47^3.7.47.2207^'**'
chacun des nouveaux facteurs des denominaleurs est egal au carr^ du. prece-
dent, diminue de deux unites ; de meme, dans la serie de Pell,
(128) l-^2 = --2 +2^73 + 2^337 + 2*.3. 17.^77 + ••• '
chaqun des nouveaux facteurs des denominateurs est egal, par les formules
de duplication, au double du carre du precedent, diminue de I'unite.
Oes developpements sont tres-rapidement convergents ; c'est, en quelque
sorte, la combinaison du calcul logarithmique et du calcul par les fractions
continues. Ainsi le denominateur de la trentieme-deuxieme fraction de la
formule (127) , est a peu pres egal a
1 /I + V5\^
V5 \~2~) '
57
226 Lucas, Theorie des FonctUyns NumSriques Simjplement P^iodiques.
et contient deux-cenf millions de chiffres^ environ ; pour ecrire le denorai-
nateur de la soixante-quatri^rae fraction de la formule (128), il faudrait
plus de deuX'Cent millions de siecUs.
Nous avons d'ailleurs demontre (Section XI), que les differents facteurs
des denominateurs sont premiers entre eux deux a deux, et contiennent, par
consequent, des facteurs premiers tons differents ; il en resulte que dans la
somme des n premiers termes de ces series, il n'y aura pas lieu de reduire
cette somme a une plus simple expression. Nous montrerons, de plus, que
tons ces facteurs, premiers et differents, appartiennent a des formes, lineaires et
quadratiques, determinees.
On a, plus generalement, Tidentite
(129) ^=1^^ + "'-''' '-^"^
(l_-z)(l_z*) ' (l_a»)(l_z«)- (i_z)(i_z«)'
si Ton remplace q par j»", on a done
(130) '-' . + ' -' '-''
(1 -z){l- z"') (1 - z"') (1 - z"'*') (1 -Z){1- ZP'*')
Si Ton fait successivement n egal a t, 2, 3, . . . n, et si I'on ajoute les egalites
obtenues, on a
n 3l^ - ^zzll _L ^'' — g^ , , z'^—z''
^ ^ (1 — Z) (1 — Z") "^ (1 — 2") (1 — 2^^) (I — 2'') (1 — 20
ZP'-Z"'*' z-z"'''
(1 _ ^;-) (1 _ z.-) (l_z)(l_,.-)
ft*'
Faisons main tenant z =, — , nous obtenons la formule
a'
^ ^ UrU^ ^ u^u,^, ^ u.^rU,./ '^'"^ "^^.v^/p-v "^ ^vtr;;^ •
On calculera d'ailleurs les numerateurs et les denominateurs de ces frac-
tions, au moyen des formules de multiplication des fonctions numeriques que
nous avons donnees. Si p designe un nombre impair, on obtient une formule
analogue en changeant U en F. On peut encore appliquer ces formules aux
fonctions circulaires.
Nous donnerons plus tard les formules analogues que Ton deduit de la
theorie des fonctions elliptiques, et, en particulier, les sommes des inverses
des termes ?7„ et de leurs puissances semblables.
Lucas, Theorie des Fonctions Nvmeriques Simplement PModiqaes, 227
Section XX.
Des relations des fonctions TJ^ et V^ avec la theorie de VSqv/ition binome.
On sait, piir la theorie de Tequation binome, exposee dans la derniere
section des Disquisitiones Arithmeticce^ que si p designe un nombre premier
impair, le quotient
Z 1 "
peut etre ecrit sous la forme
dans laquelle Yet Z sont des polynomes en z a coefficients entiers ; on prend
le signe + lorsque p designe un nombre premier de la forme 4j + 3, et le
signe — , lorsque p designe un nombre premier de la forme 4j + 1. Si I'on
fait dans cette formule z =z ^-^^ , on en deduit successivement pourp = 3, 6,
7, 11, 13, 17, 19, 23, 29, ... . , les resultats suivants :
4^1' =A[2U^+QrU;]'' + 7TV?,
(133)
2Q'^U;Y + llQ''n,
4^1^ = [2Fe, + orr^ + 4eF^ - (tl
1SQ^IV^+Q^T>
- 17^ [Fe. + Q-F^ + Q^F^ + 2(^f ,
12
4-pr = A[2U^-\- QrU,r — ^^U^ + 3(^U^ + 5(tUr-]
+ 19(2^' ir^ + QrV,r — (T F„ — 2Q^FJ^ ,
4S^=:Al2Unr+Q'U,
29r
Ur
+ 23Q* [F^ + Q'F^, — (TV^r — 2TVr-\\
+ 9Q'-]» — 29Q^ [ Vnr +(rV^—(rV^+^^V+ Q^Y,
228 Lucas, Theorie des Fonctions Nwneriques Swvplenient Periodiques.
On a, par consequent, la proposition suivante :
Theoreme : Si p designe un noinbre premier de la forme 4j + 1, fe quotient
4 —^ pent se mettre sous la forme Y^ — pZ^y et sip designe un nombre premier de
^ r
ki forme 4j + 3, ^ quotient 4 -^ pent se viettre sont la forme AY^ + pZ^ .
D'ailleurs, en changeant 2; en — z , on obtiendra un resultat semblable
pour le quotient 4 -=^ . On generalise ainsi un theoreme donne par
'^ r
Legendre, et dont la demonstration est, de cette fa§on, rendue plus simple.
II resulte encore des formules (133) une autre consequence importante. En
effet, nous avons laisse jusqu'a present A arbitraire; mais, s'il s'agit des
fonctions de troisieme espece, nous pouvons supposer — A egal au produit
d'un carre par un nombre premier p de la forme 4j + 3 ; * alors, on volt
U V
que les quotients 4 —^ et 4 — ^ sont egaux a une difference de carres, et, par
pUr pVr
suite, decomposables en un produit de deux facteurs. On a done cette
proposition :
Theoreme : Si — A est egal au produit d^un nombre premier p de la fornfie
U V
4^^* + 3 par un carri^ les quotients 4 —^ et 4 — ^ sont. quelle que soit la valeur
pUr pVr
entiere de r, decomposables en un produit de deux facteurs entiers.
Si nous considerons Tequation fondamentale
x'z=:x — 2
dans laquelle A = — 7, nous obtenons, par exemple,
Un = + 23, t/77 = — 26 4721893121 ;
et, par suite,
?7„ = — 7 X 23 X 11087 X 148303.
Nous demontrerons plus loin que les diviseurs premiers de f-jP appartien-
nent aux formes lineaires 77q ± 1; par consequent, le nombre 11087 est
premier, sans qu'il soit necessaire d'essayer ces diviseurs, puisque le premier
des nombres de la forme lineaire indiquee, est superieur a la racine carree de
11087 ; pour le facteur 148303 il n'y a que le diviseur 307 a essayer. On a
encore, dans la meme serie
U,, = — l, Un = — 384 17168 38057 ,
* £n effet, il suffit de determiner Q par la relation iQ — P^^pK*.
Lucas, Theorie des Fonctians Numerigues Simplement Periodigues. 229
et, par suite,
?7,i = — 7 X 712711 X 770041.
Ces deux derniers facteurs sont premiers; il n'y a que deux diviseurs a
essayer. On comprend ainsi comment il est possible d'appliquer le theoreme
precedent, a la recherche directe de tres-grands nombres premiers, par la con-
sideration des series de troisieme espece.
Section XXL
Sur les congruences du Triangle Arithmetigue de Pascal, et sur une generalisation
du thSoreme de Fermat.
En designant par C^ le nombre des combinaisons de m objets pris n a ti,
on a les deux formules fondamentales
^„ _ m (m — 1) • . . . (m — n + 1)
"~ 1.2.3 n
Cm /Hfn i_ /nrn — 1 ,
m ^m — 1 "T ^m — 1 )
par consequent, lorsque p est premier, on a pour n entier compris entre et ^,
la congruence
(134) ^5 = 0, (Mod.jp);
pour n compris entre et p — 1,
(135) • C5:, = (— 1)% {Uod.p);
pour n compris entre 1 et p
(136) C'; + , = 0, (Mod.p).
En d'autres termes, dans le triangle arithmetique de Pascal, tous les
nombres de la jp**^ ligne sont, pour p premier, divisibles par pj a I'exception
des coefficients extremes egaux a I'unite ; les coefficients de la (p — 1)***^ ligne
donnent alternativement pour residus + 1 et — 1 ; ceux de la (p + 1)^** ligne
sent divisibles par p, en exceptant les quatre coefficients extremes, egaux a
r unite.
Si Ton continue la formation du triangle arithmetique, en ne conservant
que les residus suivant le module jp, on reforme deux fois le triangle arith-
metique des (p — 1) premieres lignes ; puis, a partir de la (2p)""** ligne, on le
reforme trois fois ; mais les residus du triangle interm^diaire sont multiplies
par 2; a partir de la (3/>)**'^ ligne, le triangle des residus est reproduit quatre
58
230 Lucas, TMorie des Fonctions NumMques Simplement PSriodiqties.
fois, mais les nombres de ces triangles sont respectivement multiplies par les
coefficients 1, 3, 3, 1 de la troisieme puissance du binome, et ainsi de suite.
On a done, en general,
67«=Q;;XC (Mod.jp),
nil ©t Wi designant les entiers de — et de — , et u et v les residus de m et de w.
On a, de memo
C2;; = C2;^XC;7, (Mod.;)),
et, par suite,
(137) C- = C;i X C;i x Cm x . . . , (Mod.;?) ,
fii^ fi2, fisj . . . designant les residus de m et des entiers de — , —r , — ^ , . . . ,
p f f
et de meme pour ^i, ^2, 1^3, .. . .
Par consequent, si Ton veut trouver le reste de la division de CJ| par un
nombre premier, il suffit d'appliquer la formula pr6c6dente, jusqu'a ce qu'on
ait ramene les deux indices de C, a des nombres inferieurs a p.
Nous venous de voir que les coefficients de la puissance p du binome sont
entiers et divisibles par p, lorsque p designe un nombre premier, en exceptant
toutefois les coefficients des puissances p*^\ En designant par a, /3, y, .... X,
des entiers quelconques, en nombre 7?, on a done
[a + /3 + y + .... + ;^]p _ [aP + ^p + yp + .... + ;ip] = 0, (Mod. p),
et, pour am/Jziyzz =z;izzl, on obtient
vF — w = 0, (Mod. p) .
C'est dans cette congruence que consiste le theoreme de Fermat, que Ton
pent generaliser de la maniere suivante. dilferente de celle que I'on doit a
EuLER. Si a, /?, y, . . . X, designent les puissances j**'~" des racines d'une
equation a coefficients entiers, et /% leur somme, le premier membre de la
congruence precedente represente le produit par|> d*une fonction symetrique,
entiere et a coefficients entiers, des racines, et, pat* consequent, .des coefficients
de Tequation proposee. On a done
S^ = S^, (Mod.|>),
et, par Tapplication du theoreme de Fermat,
(138) S^ = S,, (Mod.^).
L'etude des diviseurs premiers de la fonction numerique S^ et de quelques
autres analogues est tres-importante ; on a, en particulier, pour n = 1 et
Si •=. 0, comme dans Tequation
or' = or + 1 ,
Lucas, TheoHe des FancUons NwnMques Simplement Periodiqties. 231
la congruence
aSp = , (Mod. p) ;
on en deduit inversement que si, dans le cos de SiZzO, an a S^ divisible par p,
pour n =1 p, et non auparavant, le nombre p est un nombre premier. En eflfet,
supposons p egal, par exemple, au produit de deux nombres premiers g et h.
On a
S,H = S,, (Mod.g)
S,^ = S,, (Mod. A);
par consequent, si Ton a trouve
S^ = 0, {Mod:ffh),
on aura aussi
S, = 0, (Mod. A),
^, = 0, (Mod. (7),
et, par le theoreme demontre,
S, = S, = 0, (Mod.gk).
Ainsi Sgf^ ne serait pas le premier des nombres S^ divisible par gh.
On pent obtenir, de cette fa^on, un grand nombre de theoremes servant,
comme celui de Wilson, a verifier les nombres premiers. Nous laisserons
de cote, pour Tinstant, les developpements curieux et nouveaux que nous
avons ainsi trouves, pour ne considerer que ceux que Ton tire des fonctions
numeriques simpleraent periodiques.
Section XXII,
Sur la thSorie des nombres premiers dans leurs rapports avec les progressions
arithmStiques,
La doctrine des nombres premiers a et6 ebauchee par Euclide et
Eratosthene. On doit a Euclide la theorie des diviseurs et des multiples
communs de deux ou plusieurs nombres donnes, la representation des nom-
bres composes a I'aide de leurs facteurs, et la demonstration de I'infinite des
nombres premiers, que Ton pent etendre facilement a la preuve de I'infinite
des nombres premiers appartenant aux formes lineaires 4a: + 3 et 6^ + 5.
Nous donnerons, dans la Section XXIV, une demonstration elementaire con-
cernant I'infinite des nombres premiers de la forme mx + 1, quelle que soit la
valeur de m. On sait d'ailleurs que, par I'emploi des series infinies, Lejeune-
Dirichlet est parvenu a demontrer I'infinite des nombres premiers de la
232 Lucas, Theorie des Functions Numeriques SimpUment Periodiques.
m
forme lineaire a + ^^i dans laquelle a et b sont deux entiers quelconques
premiers entre eux.*
On doit a Eratosthene une methode ingenieuse connue sous le nom
de Cnble Aritkmetique, qui conduit a la formation de la table des nombres
premiers et des nombres composes; on possede, depuis les travaux de Ohernac,
de BuRCKHARDT et de Dase, la table des neuf premiers millions ; Lebesque a
indique un procede qui permet de diminuer le volume de ces tables.f D'autre
part, M. Glaisher a evalue la multitude des nombres premiers compris dans
ces tables, afin de comparer les formules theoriques donnees par Gauss,
Legendre, Tchebychef et Heargrave, pour exprimer la quantite des nom-
bres premiers inferieurs a un entier donne. M. Glaisher, en comptant 1 et
2 comme premiers, a trouve les valeurs suivantes : J
pour le premier million, 78499 nombres premiers,
" deuxieme " , 70433
" troisieme " , 67885
" septieme " , 63799
" huitieme " , 63158
" neuvieme " , 62760
Les principes d'EuCLiDE et d'ERATOSTHENE conduisent ainsi a une
premiere methode de verification des nombres premiers, non compris dans
les Tables, et de decomposition des nombres tres-grands en leurs facteurs
premiers, par I'essai successif de la division d'un nombre Jixe, le nombre
donne, par tons les nombres premiers inferieurs a sa racine carree. Mais
c'est la une methode indirecte qui devient absolument impraticable, des que
le nombre donne a dix chiflfres.
En suivant cette voie, M. Dormoy est arrive par des considerations
ingenieuses, deduites de la theorie de certains nombres, qu'il a appeles
objectifs (et dans lesquels on retrouve sous le nom d'objectifs de FunitS les
diflferents term^s de la serie de Fibonacci), a Tetablissement d'une formule
generale de nombres premiers. Malheureusement, meme pour des limites
peu elevees, cette formule contient des coefficients considerables qui en
rendent Tapplication illusoire. ||
* Abhandlungen der Berliner Akademie, Berlin, 1887.
t Chernac. — Crihruyn Ari'hmeticum de I k 1020000. Deventer, 1811.
BuRCKH ARDT.— TabUs des diviseura jusqu 'A 3036000. Paris, 181 4-1 81 7.
BAQK.-^Factoren Tafeln de 6000000 k 9000000. Vienne, 1862-1866.
Lebesguk. — Tables diveraes pour la dScompoaiiion dea nombres en leurs faeieurs premiers, PariB, 1864.
X Preliminary accounts of the results of an enumeration of the primes in Dose's and BurckhardVs tabUe,
Cambridge, 1876-1877.
E. Dormoy. — Formule genhale des nombres premiers et Thiorie des Objectifs. Paris, 1867.
Lucas, Theorie des Fonctions Nunieriques Simpleinent Periodiques. 233
Les nombres premiers sont distribues fort irregulierement dans la suite
des nombres entiers ; c'est qu'en eflfet, d'une part, on voit que si ^i designe le
plus petit multiple commun des nombres 2, 3, ... m, les nombres
|M + 2, |M + 3,....,fi + m,
sont respectivement divisibles par
2, 3, .... m.
Par consequent, on pent toujours trouver m nombres consecutifs et composes,
quelle que soit la valeur de m ; mais, d'autre part, I'examen des tables permet
de constater I'existence de deux nombres impairs consecutifs, tres-grands, et
premiers. M. Glaisher a donne la liste des groupes, renfermes dans les
tables, qui contiennent au moins cinquante nombres consecutifs et composes ;
ainsi, par exemple, les suivants :
111 nombres composes et consecutifs entre 370261 et 370373,
113 " '-' " 492113 et 492227,
131 " " *' 1357201 et 1357333,
131 " " *' 1561919 et 1562051,
147 " - " " 2010733 et 2010881,
{London Mathematical Society^ 10 Mai, 1877).
On sait encore demontrer qu'uhe fonction rationnelle de n
ne pent continuellement donner des nombres premiers, puisque Ton a, quel-
que soit le nombre entier k,
^ {n + kp)=^ (n), (Mod. p),
c'est-a-dire que ^ (n) est une fonction numerique periodique d'amplitude p.
II est done fort difficile d'arriver a la loi de distribution des nombres premiers
dans la serie ordinaire des nombres entiers.
Cependant, il parait naturel d'etudier les nombres premiers d'apres
leur loi de formation. L'etude approfondie de la methode d'ERATOSTHENE
a conduit le prince A. de Polignac, a d'interessantes proprietes des suites
diatomiques ;* a la meme epoque, M. Tchebychef, arrivait par des considera-
tions peu diiferentes, a la demonstration de ce theoreme remarquable : Pour
a> 3, il y a au mains un nombre premier compris entre a et2a — 2.f On deduit
immediatement de la que le produit
1 .2.3 .... n
* Recherchea n<mvelle8 sur Us ?iombres premiers ; par M. A. de Polignac; Paris, 1851. II est curieux de
constater que, sous le nom de suite mediane^ on retrouve dans les series diatomiques, les differents termes de la
s^rie de Fsrmat.
t Journal de ZAoutfille, t. XVII.
59
234 Lucas, Theorie des FoncUons Numeriques Simplemmt PModiques.
ne saurait etre une puissance^ ni un produit de puissances^ ainsi que I'a montre M.
LiouviLLE, {Journal de Liouville^ 2* serie, t. II). En resume, ces recherches
sont basees sur la consideration des progressions arithm^tiques.
Section XXIII.
Sur la theorie des nombres premiers dans leurs rapports avec les progressions
gSomStriques.
On doit a Fermat des recherches profondes sur la theorie des nombres
premiers, et basees sur la consideration des progressions geometriqties. Cost
cette idee, distincte de la precedente, qui a donne naissance a la theorie des
residus potentieh, et, plus particulierement, a celle des residus quadratiques. De
cette fagon, on simplifie la verification des nombres premiers tres-grands, et
diviseurs de la forme a" — 1, ou plus generalement, de la forme rf* — 6", pour
a et i entiers, ainsi que la decomposition des nombres de cette forme en facteurs
premiers. Fermat avait remarque la forme Ijneaire nx + 1 des diviseurs,
et donne lui-meme la decomposition de plusieurs termes de la serie 2" — 1, et
ainsi, celle du nombre 2^ — 1, qu'il a trouve divisible par 223 [Lettre de
Fermat, du 12 Octobre 1640].
M. Genocchi a remis dernierement en lumiere un curieux passage des
oeuvres du P. Mersenne. Mais, pour en mieux saisir I'importance, nous
rappellerons en quelques mots la theorie des nombres parfaits. On dit qu'un
nombre est parfait^ lorsque il est egal a la somme de ses parties aliqtiotes, c'est-
a dire de tous ses diviseurs, excepte lui memo. En nous bornant au cas des
nombres parfaits pairs, et en designant par i, c?, des nombres premiers
differents, par n = cfb^c^d' . . . , le nombre suppose parfait, on doit avoir
2— "ft^c^ . . . = (1 + 2 + . . . + 2')(1 + 6 + J« + . . . -I- 6^)(1 -f c + c* + . . . + c») . . ,
OU bien
le second terme du premier membre est done entier, et devient, apres la
division, de la forme b^'c^' . . . ; mais d'autre part, le second membre qui contient
un nombre de termes^
^ = (/3 + l)(y + 1) . . . . ,
doit se reduire aux deux termes du premier membre ; par suite fi = 2,
/?z=l, yzz5 = = 0; done n = 2*6, et b est premier. Ainsi, les
Lucas, ThSorie des Fonctions Numeriques Siniplement Periodiques. 235
nombtes parfaits pairs appartiennent a la forme n =, 2*6, dans laquelle b doit
etre premier ; on a d'ailleurs aisement, avec cette condition
6 = 2* + ^ — 1.
En resum6, il n'y a pas d'autres nombres parfaits pairs que les nombres
dans lesquels le second facteur est un nombre premier. Cette regie etait
connue d'EucLiDE ; mais ce geometre ne savait pas demontrer que Ton obte-
nait ainsi tons les nombres parfaits pairs, sans exception.
Voici maintenant le passage des (Euvres de Mersenne :
''XIX. Ad ea quae de Nameris ad calcem prop. 20. de Ballist. & pancto 14 Praefationls ad
Hydraul. dicta sunt, adde inaentam artem qua nameri, qaotquot yolaeris, reperiantar qui cam suis
partibus aliquotis in ynicam sammam redactis, uon solum duplam rationem habeant, (quales sunt
120, minimus omnium, 672, 523776, 1476304896, & 459818240, qui ductus in 3, numerum efficit
1379454720, cuius partes aliquotse triplse sunt, quales etiam sequentes 30240, 32760, 28569920, &
alij infiniti, de quibus yideatur Harmonia nostra, in qua 14182439040, & alij suarum partium ali-
quotarum subqnadrupli) sed etiam sint in ratione data cum suis partibus aliquotis.
'* Sunt etiam alij numeri, quos yocant amicabiles, quod habeant partes aliquotas a quibus mutu6
reficiantur, quales sunt omnium minimi 220, & 284 ; huins enim aliquotse partes ilium efficiunt,
yic^que yersa partes illius aliquotse hunc perfects restituunt. Quales & 18416 & 17296; nee non
9437036, & 4363584 reperies, aliosque innumeros.
" Ybi fuerit operas pretium aduertere XXVIII numeros k Petro Bungo pro perfectis cxhibitos,
capite XXVI 11, libri de Numeris, non esse omnes Perfectos, quippe 20 sunt imperfecti, adeo vt
solos octo perfectos habeat yidelicet 6. 28. 496. 8128 33550336. 8589869056. 137438691328, &
2305843008139952128 ; qui sunt e regione tabulae Bungi, 1, 2, 3, 4, 8, 10, 12, & 29 : quique soli
perfect! sunt, vt qui Bungum habuerint, errori medicinam faciant.
'*Porr6 numeri perfecti adeo rari sunt, yt yndeeim dumtaxat potuerint hactenns inueniri : hoc
est, alii tres k Bougianis differentes : neque enim vllus est alius perfectus ab illis octo, nisi superes
ezponentem numerum 62, progressionis duplae ab 1 incipientis. Nonus enim perfectus est potesta8
ezponentis 6S minus 1. Decimus, potestas exponentis 128, minus 1. Vndecimus denique, potestas
258, minus 1, hoc est potestas 257, ynitate decurtata, multiplicata per potestatem 256.
'' Qui yndeeim alios repererit, nouerit se anoljsim omnem, quae fuerit bactenus, superasse :
memineritque interea nullum esse perfectum a 1 7 OOO potest ate ad 32000; & nullum potestatum
interuallum tantum assignari posse, quin detur illud absque perfectis. Verbi gratia, si fuerit expo-
nens 1050000, nnllus erit numerus progressionis daplas ysque ad 2090000, qui perfectis nameris
serviat, hoc est qui minor ynitate, primus ezistat.
"Ynde clarum est qukm rari sint perfecti numeri, & qokm meritd yiris perfectis comparentur;
esseque ynam ex maximis totius Matheseos difficultatibus, praescriptam numerorum perfectorum
multitudinum exhibere; quemadmodum & agnoscere num dati numeri 15, aut 20 caracteribns con-
stantes, sint primi necne, ciim nequidem sasculum integrum huic examini,*quocnmque modo hactenns
cognito, sufficiat."*
♦F. Marini Mersbnni Minimi, Coqitata Physico-Mathematica. In quibus tarn natures qudm arlis
effectus admirandi ceriissimis demonstrationibus explieantur, Paris, 1644. f° 11. de la Preface.
236 Lucas, TMorie des Fanctians NumSriques Simplenient Periodiques.
D'apres ce passage, le tableau des nombres parfaits pairs serait le suivant:
Premier nombre parfait 2 (2^-
— 1) ,
Deuxieme nombre parfait
2^(2^-
— 1),
Troisieme
2^(2^^-
— 1) ,
Quatrieme
((
2«(2'-
— 1),
Cinquieiue
u 2*2 (2^3 _
— 1) ,
Sixieme
((
2" (2" -
— 1),
Septieme
a 2*8 (2*9 _
— ij ,
Huitieme
((
2^(2^*-
— 1),
Neuvieme
a 2^(2*7_
— 1) ,
Dixierae
2126 (2127 _
- 1),
Onzieme
(( 2^^ (2^'*'^ —
— 1) .
• • • •
• • •
• •
Ce passage est d'ailleurs rapporte dans un memoire de C. N. Winsheim,
inesre dans les Novi Commentdrii Academice Petropolitance^ ad annum mdccxlix
(torn. II, pag. 78), et precede des reflexions suivantes :
"Suspicio enim adesse videtur, utrum numerus nonns, perfecti locum
tueri possit, quoniam ab acutissimo Mersenno exclusus reperitur, qui ejus in
locum potestatem binarii (2^*' — 1) 2^ sive numerum decimum nonum perfec-
turn Hanschii 1 1 47573 1 95258 1 96764 1 12927, substituit: digna carte mihi
visa sunt verba viri perspicacissimi, ut hie Integra exhibeantur."
Ainsi Me;rsenne aurait deraontre que, pour n compris entre 31 et 257, il
n'existe pas de nombres premiers de la forme 2** — 1, en exceptant ceux pour
lesquels n a pour valeur Tun des nombres
31, 67, 127, 257.
La preuve de non-decomposition du premier de ces nombres, ^^ — 1, n'a ete
donnee que plus tard, par Euler. En outre, M. F. Landry, au moyen d'une
methode inedite, et probablement fort simple, est parvenu a la decomposition
de certains grands nombres en leurs facteurs premiers ; il a, en effet, donne
la decomposition des nombres
2^1—1, 2^—1, 2*7_i^ 2^—1, 2^^ — 1,
en leurs facteurs premiers. De plus, on trouve que 2^* — 1, 2'® — 1 et 2"* — 1,
sont respectivement divisibles par 439, 2687 et 3391. Enfin, on a le theoreme
suivant :
Theoreme : Si 42 + 3 ^t 8 j + 7 sont des nombres "premiers^ le nombre
2'^^^— 1 est divisible par 8q + 7.
En efl^et, d'apres le thoereme de Fermat, on a
2«^ + «_l = 0, (Mod. 82 + 7),
et, par suite Tun des deux facteurs 2^ + ^+1 ou 2*^ + ^ — 1 du premier
membre de la congruence est divisible par le module ; mais, d'autre part,
on sait que 2 est residu quadratique de tons les nombres premiers de Tune
des formes 8w + 1 et 8w + 7 ; par consequent^ on a
2*^ + ^— 1 = 0, (Mod. 82 + 7) ;
Lucas, Thearie des Fonctions NuniSriqaes Simpleinent Periodiques. 237
en consultant la table des nombres premiers, on en conclut que pour les
valeurs de n success! vement egales a
11, 23, 83, 131, 179, 191, 239, 251, 359, 419, 431, 443, 491,
les nombres 2^ — 1 sent respectivement divisibles par les facteurs
23, 47, 167, 263, 359, 383, 479, 503, 719, 839, 863, 887, 983.
II resulte de ces diverses considerations que M£RS£NNE etait en posses-
sion d'une methode arithmetique qui ne nous est point parvenue. Cependant,
il parait naturel de penser que cette methode ne devait pas s'eloigner des
principes de Fermat, et par consequent, ne pas differer essentiellement de
celle que nous deduirerons, plus loin, de I'inversion du theoreme de Fermat.
Nous indiquons, en eflfet, comment il est possible d'arriver rapidement a
Tetude du mode de composition des grands nombres dont il est parle
plus haut.
Nous donnons dans le tableau suivant, la decomposition des nombres [7„
et Vn de la serie de Fermat, pour toutes les valeurs de n jusqu'a 64.
Parmi les grands nombres premiers de ce tableau, on remarquera
p.
Cinq nombres de dix chiffres
-
42782 55361
facteur de
2« + l,
88314 18697
((
2*' + l,
29315 42417
((
2*^+1,
18247 26041
((
2» + l,
45622 84561
i(
2«'+l;
2°. Deux nombres de onze chiffres
5 44109 72897 facteur de 2»« + 1 ,
7 71586 73929 " 2«« + 1 ;
3°. Un nombi'e de douze chiffres
16 576:5 37521 facteur de 2*' + 1 ;
4°. Quatre nombres de treize chiffres
293 20310 07403 facteur de 2« + 1 ,
443 26767 98593 " 2« — 1,
436 39531 27297 " 2*» + 1 ,
320 34317 80337 " 2^ — 1 ;
60
238 Lucas, Thearie des Fonctions Numeriques Sinvplenient PSriodiques.
5®. Un nombre de quatorze chiffres
2805 98107 62433 facteur de 2^ + 1 .
II reste a determiner la nature des trois nombres 2^^ — 1 , -^ (2*^ + 1) ,
et 2^ + 1- M. Landry pense que ces nombres sont premiers ; mais, d'autre
part, d'apres Mersenne, le premier de ces nombres serait compose ; de plus,
par la consideration de calculs que j'ai effectues, et dont la theorie est indiquee
plus loin, le dernier de ces nombres serait aussi compose. II n'y a done pas
lieu de se prononcer pour le moment.
En dehors des decompositions renfermees dans le tableau, M. Landry a
encore obtenu les diviseurs propres d'un certain nombre d'autres termes de
cette serie, a savoir
Pour 2^ + 1 4 09891 et 76 23851 ,
2^^ + 1 16 87499 65921 , (premier) ,
2^5 _ 1 I 00801 et 105 67201 ,
2'' + 1 113 38367 30401 , (premier) ,
2^^ + 1 6 64441 et 15 64921 .
*
De son cote, M. Le Lasseur est parvenu aux memes resultats; mais, il
a, en outre, indique I'identite
2*" + * + 1 z= (2^" + ^ + 2" + ^ + 1)(2^ + ^ — 2^ + ^ + 1),
qui permet d'abreger les calculs. Cette identite, fort importante, sera gener-
alisee ulterieurement.
Lticas, Thdorie des Fonctions NumSriques Simplem&it Periodiques. 239
Tableau dgs facteubs frehiebs de la s^kie BicnnBENTE de Ferhat.
ria M. F. Landbt.
u.
Diviseurs de U„
2'
Valeura de 2"
Y.
Diviseurs de T'.
S'-l
1
a
2' +1
3
V
*
3' + 1
6
»"-i
I
2"
8
2' + 1
S-
2*
16
3' + 1
17
S'-l
81
3'
32
2' +1
3.11
2"
64
2- + 1
6.13
V—l
m
8'
138 2- + 1
266 3' + 1
3.43
3'
257
v—\
1.13_
8'
512 8' + 1
3'. 19
3"
1024
2"+ 1
.5'.41
2"-l
83.89
3"
3048
2"+ 1
3.6S3
8"
4096
2-+1
17.241
!»-l
8191
3"
8193
a" + i
3.2731
3"
16384
2"+ 1
5.29.113
S"-l
7.81.161
8"
32768
2"+ 1
3M1.831
3"
65536
2"+ 1
66537
!"-l
181011
2°
131073
2"+ 1
3.43691
2"
362144
3"+ 1
5.13.37.109
S"-l
62138T
8"
524288,' 8" -1- 1
3.174763
8"
1048576! ;2" + I
17.61681
2°— 1
1M37.S3J
2"
2097152' 2" -f l'3'.43.54I9
2-
4194304 2" + l'5.397.2118
8--1
(7.178)81
8"
8388608 2"+ l'3.2796208
3-
16777216 2"+ l|97.257.673
!"-:
31.601.1301
2"
33554432V + l'3.11.251.4051
2-
67108864, 2«-f 16.53.157.1613
a"—!
7.73.262657
2"
1342177382" + l'3M9.87211
2"
268435456 |2" + i:i7.16790321
S-_l
883.1103.3089
2"
536370913: 2" 4- 13.59.3033169
2"
1073741834 2" + I 5M3.4I. 61.1321
2"-I
3147483647
2"
2147483648' 2" -f 1 3.715887883
2"
4294967296
3"+ 1 641.6700417
240 Lucas, TMorie de$ Fonctiona Num^ngue» Simplement P4riodigues,
Tableau des facteubs frehiebs de la s^rie b^curbbntb de Ferhat.
{Suite.)
u.
Diviseura de U„
8"
Valeura de 2"
V„
Diviseura de T,
a-— 1
T.S8.89.699479
8589934592
3-+ I
3'. 67. 683.30857
2"
17179869184
2-+1
5.137,953,26317
a"— 1 31.11.127.122981
2-
34359738368
8»+ I
3.11.43.281,86171
9"
687I94T6T36
2-+ I
17.241.433.38737
9"_i;223.61G318in
2"
137438953472
3"+ I
31777.25781083
2"
274877906944
a"-f 1
5.229.457.625313
3"— lit. 79.8191.121369
2"
549755813888
2"+ 1
3'. 273 1.22366891
2"
1099511627776
8"+ 1
257.4278255361
3"— I|l33e7.164511363
2"
2199033255552
2"+ 1
3.83.8831418697
a"
439S04G511I04
3"+ I
5.I3.29.1I3.1429.14449
sr_i
431-9719 20998G3
3"
8796093022208
3"+ 1
3 2932031007403
a"
17592186044416
2"+ 1
17.353.2931542417
a«_i
7.31.78.151.631.23311
2"
35184372088832
2"+ 1
3M1. 19.331.18837001
2"
70368744177664
2«+ 1
5.277.1013.1657.30269
a"— I
23514513.13261529
%'■■
140737488355328
2"+ 1
3.283.165768537531
B"
88U749767I0656
2"+ 1
193.65537.22253377
2»_1
127.4432676798593
3"
562919953421312
2"+ 1
3.43.4363953127297
3-
1125899906842624
2"+ 1
5'.41.I0I.81OI. 268501
2"— 1
7.103.2148 1I119.131D71
2"
2251799813685248
2"+l
3'.307.2857.6529 43691
2"
450359962737049G
2"+ 1
17.858001.308761441
2"— 1
6361.69431.20394401
2"
9007199854740992
9"+ 1
3.107.28059810762433
2"
18014398509481984
2»+ 1
5.13.37.109.346241.279073
2-_I
23.31. 89.88L3I91.201961
2"
36028797018963968
2" -1- 1
3,11' 683.2971 48912491
2"
72057594037927936
3"+ 1
257.5153 54410972897
S"— 1
7.32377. 524287.1212847
2"
J44115I88075355S72
2"+ 1
3'.671. 174763.160465489
2"
288230376151711744
2"+ 1
5.I073G7639 636903681
2"— I
179951.3203431780337
2-
676460752303423*88
2"+ 1
3 2H33,3717l 1824726041
2"
115292150460fi8*e976
2»+ I
17 241.61681.4668284561
2"— 1
2"
2"
2305843009213693952
4611686018427387904
2" 4 1
3
5.5681. 8681.49477. 3847T3
a"—!
7'.73.127.337.92737. 649657
2"
9223372036854775808
B"+ 1
3*.19.43.5419.77I586739a9
2"
184467440737.09561616
2"+ 1
THE ELASTIC ARCH.
By Henry T. Eddy, Cincinnati^ 0.
It is usual in the discussion of the mathematical principles of the
inelastic arch, as an arch constructed of masonry is, in eflfect, to trace the
curve of pressures due to the loading and to the thrust of the arch, and
compare it with the configuration of the arch itself. This comparison shows,
in the clearest manner, the stability or instability of the arch, and it enables
the designer to form a ready opinion as to the effect of altering the shape of
the arch, or of changing the loading, etc.
This curve of pressures is also, as is well known, the curve of equi-
librium or catenary due to the loading and to the thrust. It is also frequently
spoken of as the curve of moments, since it is well known that its ordinates
multipled by the horizontal thrust are equal to the bending moments which
would be caused by the given loading in an elastic girder on which no hori-
zontal thrust acts, which girder is not necessarily straight
It is thus seen that the girder itself can be discussed by the help of an
equilibrium polygon having any assumed thrust; and this is the process
employed in treating the girder by the graphical method.
Now when we turn to the mathematical treatment of the elastic arch, or
curved girder of any shape, acted on by a thrust, (which is the case when the
reactions of its two supports are not parallel), we find that the discussions
heretofore given have been based on analytic considerations exclusively, and
the useful relations expressed by the equilibrium polygon are left out of view.
It is the object of this paper to point out the fundamental relationship
existing between the elastic arch or crooked girder and the equilibrium curve
due to its loading and to the thrust acting upon it. This relationship con-
stitutes the basis of a complete graphical treatment of the elastic arch, subject
to any possible conditions such as induced bending moments at its extremities
or elsewhere, or the introduction of hinge joints at arbitrary points ; and it
affords at the same time a simple conception and interpretation of the analytic
results arrived at in the usual investigation of the elastic arch.
61 241
242 Eddy, The Elastic Arch.
It is evident that the bending moment at any point of an elastic arch is
the algebraic sum of the moments of the forces and couples applied to the
arch on either side of the assumed point.
Let us first consider the bending moment at any point of the arch which
is due to a simple thrust alone. A simple thrust is induced in an arch by a
variation of temperature or any other cause by which its natural span is made
to differ from the actual distance between its points of support ; but the bend-
ing moments due to a given thrust are identical whatever causes that thrust;
whether it is a secondary effect due to the bending which the loading pro-
duces, or whether it is caused by some variation of temperature or position of
its extremities which makes the natural span of the arch differ from the actual
distance between its points of support. The word thrust is used to include
the effects of contraction as well as elongation in the arch ; the thrust in the
former case being negative.
If the arch is not fixed in direction at the points of support, then the
bending moment vanishes at these points, and the thrust acts along a line
joining them, which is not necessarily horizontal. The bending moment at
any point of the arch due to this thrust is the product of the thrust by its
arm, which arm is the perpendicular distance from the assumed point to the
line of the thrust. This product is evidently equal to the product of the hori-
zontal component of the thrust by the vertical distance from the assumed
point to the line of thrusts, as appears from an elementary consideration of
the similarity of triangles.
Hence it appears that we have in this case arrived at the following
important truth:
The neutral axis of an elastic arch is the equilibrium curve of the bending
moments due to the thrust between its supports.
And this statement applies not only to an arch having hinge joints at
its points of support, but to the elastic arch in general, as we now proceed
to show.
Any case other than that already considered may be caused by the
application of a couple at one or both of the points of support, thereby
inducing a bending moment at one or both of these points. The effect of
such a couple is not dependent upon the 'manner in which it is induced : it
may be that it is arbitrarily applied, or it may be caused, as is usually the
case, by the thrust : the bending moment, which a couple arbitrarily applied
Eddy, The^ Elastic Arch. 243
at a point of support causes, is one which uniformly decreases from its point
of application to the other point of support. Its effect is then to remove the
thrust line from the point whfere the couple is applied to a new position, such
that the vertical distance between the point of support, at which the couple is
applied, and the new thrust line, multiplied by the thrust of the arch, is equal
to the moment of the applied couple. The other extremity of the thrust line
is unmoved by applying this couple. If in addition a second couple be applied
at the second point of support the thrust line is removed from that point of
support in a similar manner while the other extremity is unmoved.
But the same reasoning now applies, which we before employed, to find
the moment at any point of the arch ; it may be stated more explicitly thus :
The bending moment at any point of an elastic arch caused hy the thrust^
horizontal or inclined^ is the product of the horizontal component of the thrust hy
the vertical ordinate between the point assumed on the neutral axis of the arch and
the thrust line in Us true position.
It may be noticed in this connection that it is always possible to deter-
mine the magnitude of the couples accompanying a simple thrust and applied
at the points of support, from consideration of the conditions imposed on the
arch by the amount of deflection horizontal or otherwise which is possible,
but it does not fall within the scope of this paper to examine these conditions
and show how to determine the couples accompanying a thrust.
Having considered the thrust and its equilibrium or moment curve, which
is the neutral axis of the arch itself, let us in the second place consider the
loading and its moment curve.
The bending moment at any assumed point of the arch due* to the weights
is the algebraic sum of the products obtained by multiplying each weight by
its horizontal distance from the assumed point.
If the arch has hinge joints at the points of support, the loading can
cause no bending moments at those points ; but, if the arch is supported in
some other way, it is evident that the moments due to the loading are accom-
panied by couples applied at the points of support, in the same manner as
were the bending moments due to the thrust, and that the magnitude of these
couples must be determined from the same considerations respecting deflec-
tion, etc., as determined those ; indeed, each separate force which is applied
to the arch, be it thrust, weight or any other force, causes bending moments
throughout the arch which can be separately treated, and each must evidently
244 Eddy, The Elastic Arch.
be treated in the same manner. We have, for convenience, grouped all the
weights together. Hence we have reached another important truth :
In any elastic arch^ the closing line of the moment curve, due to the weights
alone, must be found from the same conditions and have the same relations to this
moment curve that the thrust line has to the curved neutral aads of the arch.
It is to be noticed that the thrust caused by loading an arch induces
bending moments of opposite sign from those induced by the loading itself,
and that the bending moments really acting on the arch are the difference
between those induced by the loading and those induced by the thrust.
Hence appears the truth of the following statement, which combines together
the separate effects of the thrust and the loading :
If that moment curve, due to the loading, which has for its horizontal thrust tlie
thrust really acting in the arch, be superposed upon the curve of the arch itself in
such a manner that its closing line coincides with the thrust line of the arch, then the
bending moment at any point of the arch is equal to the product of the Iiorizontal
thrust by the vertical ordinate between the assumed point and the moment curve due
to the loading.
Hence the neutral axis of the arch may be considered to play the part of
a curved closing line of the moment curve.
This proposition, respecting the coincidence of the thrust line and the
closing line, affords the basis for a new graphical investigation of the elastic
arch *
The same principles may be applied to the elastic arch acted on by other
than vertical forces.
*8ee **New Constructions in Graphical Statics." By Henry T. Eddy, published by D. Van Nostrand,
New York, 1877.
RESEARCHES IN THE LUNAR THEORY.
By G. W. Hill, Nyack Ikimpike, N. Y.
Chapter II.
(Continued from p. 147.)
The method of employing numerical values^ from the outset, in the
equations of condition, determining the a^, is far less laborious than the
literal development of these coefficients in powers of a parameter. For com-
parison with the results just given, we add the calculation of the coefficientsr
by this method. The following table gives the numerical values of the sym-
bols |j, *], \j\ and (^■), but the division by the quantity 2 (4f — 1) — 4m -f m"
has been omitted ; it is easier to perform this once for all at the end of the
series of operations, than to divide each coefficient separately. Hence it must
be understood that all the numbers in each department of the table are to be
divided by the divisor which stands at the head of it.
Coefficients for a, and a_i .
Divisor = 5.68314 08148 64695.
[1]
(1)
[1,-2]
[1, -1]
[1,2]
0.00861 47842 96261
0.00623 66553 18347
13.30665 60411
6 32993 22853
10.71949 01593
[-
[■
[-1] = -
(-1) = -
-1, -3] = -
-1, -2] = -
[-1, 1] = -
0.01178 75756 56865
0.04941 95042 02516
66.98979 68560
28.01307 31002
10.96365 06556
[1,3]=— 15.10904 80332
t— 1,2]=— 38.67409 27816
Coefficients for a, and a_2 .
Divisor = 29.68314 08148 64695.
2 [2]
2(2)
[2, -2]
[2, -1]
[2,1]
00205 43632 76229
0.02909 07097 39048
14.97672 37558 [-
9.32666 40103 [-
13.00326 82750 49 [-
2 [-2] =
2 (-2) =
-2, -4] =
-2,-3] =
-2,-1] =
0.01834 79966 76898
0.07227 35686 23216
108.69686 46706
63.00980 48251
8.67987 25398
S2
245
Hill, Beaearckes in the Lunar Theory.
[2,3]=-
60.03961 76194
[-2,1] = -
■ 3 64362 31954
[2,4]=-
74.07269 86888
[-2,2]=-
- 19.61044 21261
Coefficients for a^ and a_a .
Diviaor = i
39.68314 08149.
[3] = -
0.00113 36729 26473
[-3]=-
- 0.00793 43696
(3) = -
0.01768 33677
(-3)=-
- 0.03207 76506
[3,-1] =
12.99224 12619
[-3,^] = -
-114.67638 20668
[3,1]=-
18.10997 74284
[-3,-2]=-
- 36.67316 33864
[3,2]=-
41.33769 10334
[-3,^1]=-
- 12.34544 97816
[3,4] = -
103.14632 67728
[-3,1] =
1.46318 69680
Coefficients for &t and a_4.
'
Divisor =
126.68314 08 .
2[4] = -
0.00428 9733
2[-^]=-
- 0.01449 0913
2(4)=-
0.03864 29168
2(^)=-
■ 006023 435
[4,-1] =
16.82602 987
[-4,-6]=-
-182.50817 069
[4,1] = -
22.66333 2
[_^,_3]=^
- 7901980 9
[4,2]=-
61.16496 6
[-4,-2]=-
- 4251817 6
[4,3]=-
86.60490 2
[-^,-1]=-
- 1617823 8
[4,6] = -
17X69968 136
[-^,1] =
601654 053
Coefficients for ag and a_, .
Divisor =
= 197.68314.
[6]=-
0.00272 9636
(6) = -
- 0.02896 299
[6,1]=-
26.99634 4
[-6,-^]^
-138.68780
[6,2]=-
60.26133 2
[_^,_3] = _
■ 89.421810
[6,3]=-
99.79796 8
[-6,-2]=-
■ 49.88518 4
[6,4]=-
146.60623 2
[-6,-1]=-
- 20.077912
Coefficients for a, and a_8 .
Divisor :
= 286.68314.
2[6]=-
0.00622 021
2(-6)=-
• 0.06640 548
[6,1]=-
31.21669
[-6,-5] = -
-214.46646
[6,2]=-
68.99224
[-6,-4]=-
-152.69091
[6,3] = -
11332666
[-6,-3]=-
-100.36648
Hill, Researches in the Lunar Theory. 247
[6, 4] = — 164.21995 [_6,— 2]=— 57.46319
[6, 5] = — 221.67212 [_6,— 1] = — 24.01103.
These numbers are arranged for carrying the precision to quantities of
the 13th order inclusive, and to 15 places of decimals. The quantities [J, i\
can be tested by differences, if and the divisor with the negative sign are
inserted in the proper places in the series of numbers ; for it is evident that
the second differences should be constant..
The final results are given below, where, in order that the degree of
convergence of this process may be appreciated, we have given the value
arising from the first approximation, and then, separately, the corrections
arising severally from the second and third approximations. It must ba borne
in mind that each of these terms is the numerical value, not of an infinite
series, but of a rational function of m, and, consequently admits of being
computed exact to the last decimal place employed, and, in fact, is here so
computed. Hence any error there may be in these values of the a< arises only
from the neglect of the terms of the following approximations, which, in half
the number of cases, are of the 14th order, and, in the other half, of the 16th
order. It is safe to aflSrm that these cannot, in any case, exceed two units
in the 15th decimal.
&i • a 1 •
1st apx , term of 2d order, + 0.00161 68491 71693 — 0.00869 68084 99634
2d " " 6th " —0.000000141698831 +0.0000000615 61932
3d " " 10th " +0.000000000006801 —0.000000000013838
s^= + 0.00161 67074 79563, a_, = — 00869 67469 61540,
a«2 • a 2 •
1st apx., term of 4th order, + 0.00000 68793 36016 + 0.00000 01636 69406
2d " " 8th " -0.0000000006 78490 +0.000000000121088
3d » " 12th " +0.000000000000052 —0.000000000000007
^ = + 0.00000 68786 66578, a_j = + 0.00000 01637 90486,
a3 .. a_ 8 •
Ist apx.,terra of 6th order, + 0.00000 00300 36769 + 0.00000 00024 60338
2d " " 10th " —0.000000000004128 +0.000000000000065
a,^ = + 0.00000 00300 31632, a_, = +0.0000000024 60393,
248 Hill, Besearches in the Lwnar Theory.
1st apx., term of 8th order, + 0.00000 00001 76296 + 0.00000 00000 12284
2d " " 12th » —0.000000000000028 0000000000000000
a, = + 0.00000 00001 76268, a_4 = +0.00000 00000 12284,
ft© ^0
Of the 10th order, % = + 0.00000 00000 01107, ^ = + 0.00000 00000 00064,
Of the 12th order, a, = + 0.00000 00000 00007, a_, = + 0.00000 00000 00000.
ao a.
These give the following numerical expression for the coordinates,
r cos i> z= ao [1 — 0.00718 00394 81977 cos 2r
+ 0.00000 60424 47064 cos 4r
+ 0.00000 00324 92024 cos 6r
+ 0.00000 00001 87662 cos 8t
+ 0.00000 00000 01171 cos lOr
+ 0.00000 00000 00008 cos 12r] ,
r sin i> = a„ [ 0.01021 14644 41102 sin 2t
+ 00000 67148 66093 sin 4ft
+ 0.00000 00276 71239 sin 6t
+ 0.00000 00001 62986 sin 8t
+ 0.00000 00000 01042 sin lOr
+ 0.00000 00000 00007 sin l^r] .
On comparison of these values with those obtained from the series in Itt, the
differences are found to be only some units in the 11th decimal.
The coefficients tend to diminish with some regularity as we advance
towards higher orders. This is shown by the following scheme of the logar-
ithms and their differences :
n 97.8661
98.0091
- 3.2621
94.7812
94.7570
+ 9366
—
- 2.2694
«
2.3165
92.6118
2.2387
+ 307
92.4405
871
2.2294
90.2731
2.2046
341
90.2111
363
2.1931
88.0686
2.1809
237
88.0180
201
2.1730
86.8876 85.8460
Hill, Besearches in the Lunur Theory.
249
For verification the following equations were computed,
2,. [(2i + 1 + m)2 + 2m^ a,. [2,. aj^ = 1.17141 84591 84518 aj ,
2, . (— ly (2e + 1) {2i + 1 + m) a, . [2, . (— 1)' a,]^ = 1.17141 84591 84513 ag .
The small difference between the numbers is explained by the fact that, in
these formulae, the quantities a^ are, when i is somewhat large, multiplied by
large numbers ; as, for instance, ag by 169. From the average of these two
results we get
ao = 0.99909 31419 75298 f-'^l * .
In the investigations of succeeding chapters, the function ^ plays an
important part. Hence we will here derive its development as a periodic
function of r by the method of special values By dividing the quadrant,
with reference to r, into 6 equal parts, we obtain the advantage that the sines
or cosines of the multiples of 2t are either rational or involve VST The
special values of the coordinates and of — , thence deduced, are
t.
r
— cos V.
&0
r .
— sin V.
0°
0.99282 60356 45842
0.00000 00000 00000
15
0.99378 49245 37167
0.00511 07041 52675
30
0.99640 69264 50272
0.00884 83280 32746
45
0.99999 39577 40480
0.01021 14268 70906
60
1.00358 70309 15127
0.00883 84298 76613
75
1.00622 11177 22330
0.00510 08054 31947
90
1.00718 60496 23406
0.00000 00000 00000
1.19699 57017 23421
1.19348 68051 03032
1.18399 66676 76716
1.17125 64904 33157
1.15876 77987 29687
1.14978 07679 95764
1.14652 34925 50570 .
From the numbers of the last column, by the known process, we deduce
-, = 1.17150 80211 79225
-f 0.02523 36924 97860 cos 2t
+ 0.00025 15533 50012 cos 4t
+ 0.00000 24118 79799 cos 6r
+ 0.00000 00226 05851 cos 8t
+ 0.00000 00002 08750 cos IOt
+ 0.00000 00000 01908 cos 12t
+ 0.00000 00000 00017 cos 14t .
6S
250 Hill, Researches in the Lunar Theory.
The last coefficient has been added from induction, after which it becomes
necessary, as is plain, to subtract an equal quantity from the coefficient of
cos lOr. Writing the logarithms, as in the former case, we have, the last
logarithm being supplied from estimation,
A A^ A^
98.4020
- 2.0014
96.4006
-168
2.0182
4-68
94.3824
100
2.0282
36
92.3542
64
2.0346
20
90.3196 ,
44
2.0390
10
88.2806
2.0424
34
86.2382
It will be noticed how much slower this series converges than those for the
coordinates.
Any information regarding the motion of satellites having long periods
of revolution about their primaries will doubtless be welcome, as the series
given by previous investigators are inadequate for showing anything in this
direction. Hence this chapter will be terminated by a table of the more
salient properties of the class of satellites having the radius vector at a mini-
mum in syzygies and at a maximum in quadratures. For this end I have
selected, besides the earth's moon, taken for the sake of comparison, the
moons of 10, 9, 8, ... , 3 lunations in the periods of their primaries, and also
what may be called the moon of maximum lunation, as, of the class of satel-
lites under discussion, exhibiting the complete round of phases, it has the
longest lunation.
In order that the table may be readily applicable to satellites accom-
panying any planet, the canonical linear and temporal units, that is those
for which ^ and n' are both unity, will be used.
From the foregping methods we obtain :
r COS i> = a [1 — 0.011230 cos 2t r sin v = a [ 0.016102 sin 2v
+ 0.000015 cos 4t] , + 0.000014 sin 4t] ,
log a = 9.3051648 .
Hill, Researches in the Lunar Theory. 251
For m =
9 '
r cos r = a [1 — 0.014044 cos 2t r sin u = a [ 0,020232 sin 2r
+ 0.0000247 cos 4r] , + 0.0000230 sin 4t] ,
log a = 9.3326467 .
For m = -Q- ;
r COS u = a [1 — 0.018061 cos 2t r sin i; = a [ 0.026172 sin 2t
+ 0.0000421 cos 4t + 0.0000388 sin 4t
+ 0.00000057 cos 6r] , + OJ00000048 sin 6r] ,
log a = 9.3630019 .
For in = -=- ;
7
r cos w = a [1 — 0.02407886 cos 2t r sin d = a [ 0.03516059 sin 2r
+ 0.00007760 cos 4t + 0.00007063 sin 4t "
+ 0.00000141 cos 6t + 0.00000118 sin &r
+ 0.000000025 cos 8t] , + 0.000000022 sin 8t] ,
log a = 9.3969048 .
For m = -jr- ;
r cos D = a [1 — 0.03368245 cos 2t r sin u = a [ 0.04968194 sin 2t
+ 0.00015943 cos 4ir + 0.00014312 sin 4t
+ 0.000004077 cos 6r + 0.000003393 sin 6t
+ 0.000000097 cos 8r] , + 0.000000084 sin 8t] ,
log a = 9 4352928 .
For m = -=- ;
o
r cos r = a [1 — 0.05038803 cos 2t r sin d = a [ 0.07536021 sin &-
+ 0.00038127 cos 4t + 0.00033582 sin 4t
+ 0.000014686 cos &r + 0.000012168 sin &r
+ 0.000000505 cos 8r] , + 0.000000438 sin Br] ,
log a = 9.4795445 .
262 Hill, Researches in the Lunar Theory.
-Form = i-,
4
r cos i; = a [1 — 0.08331972 cos 2t r sin d = a [ 0.12709553 sin 2^
+ 0.00114564 cos 4t + 0.00098090 sin 4t
+ 0.00007409 cos 6r + 0.00006099 sin 6r
+ 0.00000404 cos 8t] , + 0.00000342 sin 8t] .
log a = 9.5318013. .
For m = — ;
r 90s i; = a [1 — 0.1622330 cos 2t r sin i; = a [ 0.2542740 sin 2t
+ 0.0048920 cos 4r + 0.0039840 sin 4t
+ 0.00059858 cos 6r + 0.00049306 sin 6t
+ 0.000081198 cos 8t + O.CO(X)70196 sin 8r
+ 0.000011873 cos lOr + 0.000010611 sin lOr
+ 0.000001849 cos 12t] , + 0.0000016902 sin 12r] ,
log a = 9.5955815.
For moons of much longer lunations the methods hitherto used are not
practi6able, and, in consequence, we resort to mechanical quadratures. Here
we shall have two cases. The satellite may be started at right angles to and
from a point on the line of syzygies, and the motion traced across the first
quadrant ; or it may be started at right angles to and from a point on the
line of quadratures, and the motion traced across the second quadrant ; the
prime object being to discover what value of the initial velocity will make
the satellite intersect perpendicularly the axis at the farther side of the
quadrant.
The differential equations
give, as expressions of the values of the coordinates, in the first case,
a: = xo + ^fjdt-f/yi, - 3] .dt^ ,
y = 2J^{.,-x)dt-CC^-dt
Hill, Researches in the Lunar Theory. 253
and, in the second case,
Here the subscript (0) denotes values which belong to the beginning of
motion, and d) will hereafter be used to denote those which belong to the end.
Let V be the velocity, and a the angle, the direction of motion, relative
to the rotating axes, makes with the moving line of syzygies. In the first
case then a = 90°, and we wish to ascertain what value of Vq will make
(Ti = 180°. Generally, for small values of Vq , Ci will come out but little less
than 270° ; but, as Vq augments, a^ will be found to diminish, and, if Xq does
not exceed a certain limit, a value of Vq can be found which will make
(Ti = 180°. In the second case, in like manner, we seek what value of Vq
will make a, = 270°.
Mechanical quadratures performed with axes of coordinates having no
rotation possess some advantages, as, in this case, the velocities are not
present in the expressions of the second diflferentials of the coordinates.
Let X and Y denote the coordinates of the moon in this system, and X
its longitude measured from the line of the last syzygy, from which t is also
counted. Then the potential function is
n = — — 4-/^ + 4 (-^ cos ^+ FsinO'.
r 2 2 ^ ^
And
Therefore, if we compute p and from
jp cos d = — — — 2r\ cos (;i — t) ,
p sin z= — I — + r sin (;i — t) j
we shall have
-^^=i>cos {e + t),
iPY
^^ =P sin (fi + •
64
254 IIiLL, Researches in the Lunar Theory.
The needed values of v and a can be derived from the equations
V cos (a + ^) = --- + F,
V sin (c + zz — — — X.
The developments of the coordinates in ascending powers of ty t being
counted from any desired epoch, can often be employed with advantage.
Differentiating the differential equations n times we have
cir^^ ~ dr^' '^ dt^ rfr ^ ^'
^ zz — 2 {r~ y) .
Also
with a similar formula for the differential coefficients of r~'^.
The differential coefficients of r~*, as far as the 4th, are •
dt \ dt^^ dt)'
dUj^') o^sC d'x , dh , dx" , df\ , .._:/ dx , (M*
dt' ~ \ df^^dt'^ dt dt'^ dt dt'J
+3<v-'(4:+4)(4;?+4^+s+D
4- 30r-' Ta- '^'^ 4- « '^'^ 4- '^ 4- '^A'
Hill, Besearches in the Lunar Theory. 255
By means of these formulae x and y can be expanded in series of ascending
powers of t, as far as the term involving f, provided we know the values of
Xj y, — and -^- corresponding to ^zz . Taking t suflSciently small to make the
uZ uZ
terms, involving higher powers of t than the sixth, insignificant, as, for instance,
t = 0.05 or ^ = 0.1 , we can ascertain the values of x, y, -^ and ^ at the end
^ dt dt
of this time. With these values we can again construct new series for x and
y in powers of <, in which the latter variable is counted from the end of the
previous time. By repetitions of this process the integration can be carried
as far as desired. Jacobi's integral, which has not been put to use in the
preceding formulae, can be employed as a check.
In case the body starts from, and at right' angles to, either axis, the
coeflScients of every other power of t in the series for the coordinates vanish.
Thus when the axis in question is that of x^ the series for the coordinates
have the forms
y = Vot + A,t' + A,t' + A-;f + A,t' +
By substitution of these values in the differential equations and the equating
of each resulting coeflScient to zero we arrive at the following equations ;
1.2A^ = 2vq + ^Xq — Xq^ ,
2 . 3 ^3 = — 4^2 — ^V\ 1
3AA, = 6A, + 3A, + \ x^' {3vl + 4xoA,) ,
4.5 As = — 8^4 + Y x^\ {vl + 20:0^2) — ^o^A^ ,
5.6A, = lOA, + 3A, + I XV' {^v^A^ + 4xoA, + 3AI)
— ^ x^^ {vl + 2a:o ^2) {^l + €^0^2) ,
6.7 A, = — 12 A, + 1 xv\ {2voA, + 2xoA, + A^ — J a:v\ K + 20:0^*2)'
+ -^ x^ [vq 4" 2xqA2) A^ — x^ As J
7.8As= UA, + 3^6 + 1 ^r' (61^0^5 + ^oA, + QA^A, + Al
— I ^o" ' K + 2^:0^2) (10t;o^3 + 8^o^4 + ^?)
16
2"
+ Y ori"^ {2vqA^ + 2xqA^ + ^i) A^ ,
256 Hill, Researches in the Lunar Theory.
8.9^,= - \QA, + 1 x^\ {2v,A, + 2x,A, + 2A,A, + ^I)
— T ^V'^l (^'o + 2x,A,) {2v,A, + 2x,A, + A\)
+ g xv\ {< + 2xoA,y + I a:^« (21^0^3 + 2x,A4 + Al) A,
— y ^^' {vl + 2xoA,y A, + I- XV' {vl + 2xoA,) A, — a^'A^ .
By means of these relations each A can be derived from all the A which
precede it.
When the axis is that of y, the series have the forms
X = vot + A,t^ + A,f + A^f + A,P + . . . ,
t/ = i/o + A,t' + A,t' + A,t' + A,t' + ....
And the equations, determining the coefficients -4, are
1.2^, = — 2^0 — 3^%
2 . 3 -/ig = 4^2 + 3^0 — pV\ ,
3AA, = — 6A, + \yv' {Zvl + 4.y,A,) ,
4.5^ = 8^4 + 3^ + Jyir' V, {vl + 2y,A^) —y^'A,.
The equations are not written as far as in the former case, as it is evident
they may be derived from the preceding group by putting y^ in the place of
Xq^ reversing the signs of the first terms, and removing the term 3^»_2 from
the equations, which give the values of the A of even subscripts, into those
which give the values of the A of odd subscripts, after having augmented the
subscript by unity.
The velocity of the moon of maximum lunation vanishes in quadratures,
and when Vq =, the preceding series become, putting yj"^ = a ,
^ V18I44O 12096 ^45360 ^ 9072 /
/ 1 317 a_ 13 3 10403 , 947 A iH
"^ V19958400""^ 9979200* 120960" 4989.600**"^ 237600* /J'
. /" 1 1 2 . 13 3 73 A ^
"*'V10080" 1008* "^1120* "5040*/
^v 907200 ^90720 756 ^ 453600 400 / J "
Hill, Besearches in the. Lunar Theory. 257
These series suflSce for computing the values of x and y with the desired
exactitude when t is less than 0.3.
This special case of the moon of maximum lunation will now be treated.
As there seems to be no ready method of getting even a roughly approximate
value of yo) we are reduced to making a series of guesses. I first took y^ •=, 0.82 ;
tracing the path to its intersection with the axis of a: , Ci , which ought to be
270°, came out 261° 29^ 47^.9. A second trial was made with y^ = 0.7937 ;
the result was a^ = 267° 37' 8".3 . Again a third trial with yo = 0.7835 gave
Ci = 269° 4r 13\3. The principal data acquired in the three trials are given
in the following lines :
ni T /r ^ y^ /* Maximum
3^0 • ^ • ^1 • ^ • ^ • ^i • Variation.
0.8200 0.972430 —0.339523 —0.288149 —1.927275 261° 29' 47".9 44° 57' 4''
0.7937 0.908207 —0.290945 —0.089184 —2.144832 267 37 8 .3 46 39 36
0.7836 0.884782 —0.274324 —0,012170 —2.227928 269 41 13 .3 47 17 21.
1 denotes the time employed in crossing the quadrant, and the last column
contains the maximum value of the angular deviation of the body from its
mean direction as seen from the origin, that is, the direction it would have
had, had it moved across the quadrant with a uniform angular velocity about
the origin.
A check may be had on the accuracy of the computations by mechanical
quadratures. We determine the value of the constant 2(7 which completes
Jacobi's integral from the coordinates and velocities, both at the beginning
and at the end of the motion, for each of the three trials. The result is
y^ , First value. Second value.
0.8200 2.34902 2.43901
0.7937 2.51985 2.51987
0.7835 2.55265 2.55261 .
We can now apply Lagrange's general interpolation formula to these
data, and, regarding (Tj as the independent variable, inquire what are the
values which correspond to a^ = 270°. The numbers of the first trial must
be multiplied by +0.014861; those of the second by —0 210190; those of
the third by + 1.195329 , and the sums taken. The results are
mg T ^ ^1 ^^1 O/^ Maximum
^ dt dt Variation.
0.781898 0.881160 0.271798 —0.000083 —2.24093 2.55788 47° 23' 12".
65
258 Hill, Researches in the Lunar Theory.
dx • • ••
That — — does not rigorously vanish is due to the employment of only three
terms in the interpolation ; for the same reason the value of 2C does not quite
agree with that obtained from the values of x^ and -~ . To make all these
at
elements accordant we add 0.00009 to the value of -^ .
at
A table of approximate values of x and y, derived roughly from the data
aflfbrded by the process of mechanical quadratures is appended: they will
serve for plotting the orbit.
t.
.r .
y- ^
t.
X .
y-
t.
X .
y-
0.00
.0000
+ .7819
0.30
— .0148
+ .7080
0.60
— .1177
+ .4748
0.02
.0000
.7816
0.32
.0180
.6978
0.62
.1294
.4519
0.04
.0000
.7806
0.34
.0215
.6869
0.64
.1418
.4277
0.06
.0001
.7790
0.36
.0256
.6752
0.66
.1547
.4022
0.08
.0003
.7767
0.38
.0301
.6629
0.68
.1680
.3762
0.10
.0006
.7737
0.40
.0361
.6499
0.70
.1818
.3466
0.12
.0009
.7701
0.42
.0407
.6361
0.72
.1956
.3162
0.14
.0015
.7659
0.44
.0468
.6216
0.74
.2095
.2839
0.16
.0022
.7610
0.46
.0534
.6063
0.76
.2230
.2496
0.18
.0032
.7554
0.48
.0607
.6902
0.78
.2369
.2131
0.20
•
.0044
.7492
0.50
.0686
.5733
0.80
.2476
.1746
0.22
.0058
.7432
0.52
.0771
.5656
0.82
.2575
.1839
0.24
.0076
.7347
0.54
.0863
.6369
0.84
.2663
.0913
0.26
.0096
.7265
0.56
.0961
.5172
0.86
.2704
.0474
0.28
.0120
.7176
0.68
.1066
.4965
0.88
.2718
.0027
The following is the table of the numerical values of the quantities of
principal interest belonging to the moons mentioned at the beginning of
this paragraph. In the first line stands the earth's moon, having very
approximately 127% lunations in the period of its primary. In the last line
is the moon of maximum lunation. The quantities belonging to the moon of
two lunations have been somewhat rudely inferred from the numbers in the
adjacent lines.
Hill, Researches in the Lunar Theory.
259
N umber of
Lunations
in period
of Primary.
Kaditts
Vector
in
Syzygieg.
Radius
Vector
in Quad-
ratures.
Ratio.
1
Velocity
in
Syzygies.
Velocity
in Quad-
ratures.
Ratio.
2C,
Maximum
Variation.
1
m
^0-
n-
t>o.
t),.
121^,^
0.17610
0.17864
1.01446
2.22295
2.16484
0.97386
6.50888
0°
35' 6"
10
0.19965
0.20418
1.02271
2.06163
1.97693
0.95892
6.88686
56 21
9
0.21209
0.21813
1.02849
1 .98730
1.88501
0.94853
6.61662
1
9 33
8
0.22652
0.23486
1.03678
1.90904
1.78250
0.93372
5.33873
1
29 68
7
0.24S42
0.25543
1.04934
1.82721
1.66572
0.91162
5.06535
2
63
6
0.26332
0.28167
1.06969
1.74333
1.52851
0.87677
4.76409
2
60 49
5
0.28660
0.31699
1.10605
1.66247
1.35953
0.81777
4.46103
4
18 37
4
0.31232
0.36897
1.18138
1.60111
1.13480
0.70876
4.13277
7
17
3
0.33235
0.45973
1.38329
1.62141
0.79387
0.48962
3.72018
14
34 14
2
0.302
0.684
2.26
2.00
0.18
0.09
2.89
37
21
1.78265
0.27180
0.78190
2.87676
2.24102
0.00000
0.00000
2.55788
47
23 12
In regard to this table we may notice the following points. The moon of
the last line is the most remarkable : it is, of the class of satellites considered
in this chapter, (viz., those which have the radius vector at a minimum in
syzygies, and at a maximum in quadratures,) that which, having the longest
lunation, is still able to appear at all angles with the sun, and thus undergo
all possible phases. Whether this class of satellites is properly to be pro-
longed beyond this moon, can only be decided by further employment of
mechanical quadratures. But it is at least certain that the orbits, if they do
exist, do not intersect the line of quadratures, and that the moons describing
them would make oscillations to and fro, never departing as much as 90°
from the point of conjunction or of opposition.
This moon is also remarkable for becoming stationary with respect to the
sun when in quadrature ; and its angular motion near this point is so nearly
equal to that of the sun that, for about one-third of its lunation, it is within 1°
of quadrature. From the data of the table we learn that such a moon, circu-
lating about the earth, would make a lunation in 204.896 days.
We notice that the radius vector in syzygies of this class of satellites
arrives at a maximum before we reach the moon of maximum lunation. This
260 Hill, Researches in the Lunar Theory.
maxiraum value is very nearly, if not exactly, J, when measured in terms of
our linear unit, and thus is a little less than double the radius vector of the
earth's moon. It occurs in the case of the moon which has about 2.8 luna-
tions in the period of its primary.
The radius vector in quadratures augments continuously as the length of
the lunation increases, as also does the ratio of these radii, until, in the moon
of maximum lunation, the radius in quadratures is but little less than three
times that in syzygies.
The velocity in syzygies does not continuously diminish, but attains a
minimum somewhere about the moon of four lunations, and afterwards aug-
ments so that, for the moon of maximum lunation, it does not differ greatly
from the velocity of the earth's moon in syzygies. On the other hand the
velocity in quadratures constantly diminishes.
The maximum value of the variation augments rapidly with increase in
the length of lunation, so that, in the moon of maximum lunation, it exceeds
an octant, or is more than 80 times the value which belongs to the earth's
moon.
In the adjoining figure are constructed graphically the paths of the
earth's moon, of the moons of four and three lunations, and of the moon of
maximum lunation. The moons in the first lines of the table have paths
which approach the ellipse quite closely, but the paths of the moons of the
last lines exhibit considerable deviation from this curve, while the orbit of
the moon of maximum lunation has sharp cusps at the points of quadrature.
( To be continued. )
Hill, Researches in the Lunar Thewy.
BIBLIOGRAPHY OF HYPER-SPACE AND NON-EUCLIDEAN
GEOMETRY.
By George J5ruce Halsted, Tutor in Princeton College, N, J.
Until the present century the Euclidean Geometry was supposed to be
the only possible form of Space-science; that is, the space analysed in
Euclid's axioms was supposed to be the only nou -contradictory sort of space.
The Parallel-Postulate was generally supposed to be a consequence of
the nature of straight lines, and demonstrable from the remaining postulates
and axioms. The researches of Peyrard show that it was not given out by
Euclid as an axiom, since in all the MSS. examined by him it is kept
separate from the axioms, and has only been classed with them by an obvious
error in modern times. The enormous number of unsatisfactory attempts
to prove this postulate, led finally to a systematic development of the results
obtainable when it is denied, and then sprang forth the Non- Euclidean or
Absolute Geometry.
In the " Encyclopsedie der Wissenschaften und Kiinste ; von Ersch und
Gruber; Leipzig, 1838; under "Parallel," Sohncke says that in mathematics
there is nothing over which so much has been spoken, written and striven,
as over the theory of parallels, and all, so far, (up to his time) without
reaching a definite result and decision. He divides the attempts into three
classes: — 1. In which is taken a new definition of parallels. 2. In which is
taken a new axiom different from Euclid's. But just as Euclid's cannot be
considered axiomatic, so is it with these new postulates. This led to the third,
the largest and most desperate class of attempts, namely, to deduce the
theory of parallels from reasonings about the nature of the straight line and
plane angle. The article is followed by a carefully prepared list of ninety-
two authors on the subject, from the earliest times up to the year 1837.
In English an account of like attempts is given in the ''Geometry without
Axioms," by Perronet Thompson : Cambridge, 1833 ; where the author also
makes an elaborate attempt of his own. These accounts may be considered
to bring the subject up to the point where, through the perfectly original
66 261
262 Halsted, Bibliography of Hyper-Space and Non- Euclidean Geometry.
works of two new geometers, it assumed a totally new aspect and became the
question of Non-Euclidean Geometry. At this point we take up its Biblio-
graphy, together with that of Hyper-Space, which, though springing at first
from a purely analytical basis, has become intimately connected with the
former.
«
* 1. LOBATCHEWSKY, NlCOLAUS IVANOVITCH. (1793-1856).
The first public expression of his discoveries was given in a discourse at
Kasan, February 12, 1826.
I. Principien der Geometrie. Kasan, 1829-30.
II. Ifeue Anfangsgruende der Geometrie, mit einer vollstandigen Theorie
der Parallelen. Gelehrte Schriften der Universitat Kasan, 1836-38. His
chief work (orig. pub. in Russian). Hoiiel has made, a translation of it into
French (in MS.)
III. Geometrie Imaginaire. Crelle's Journal, B. XVII, pp. 295-320.
1837.
IV. Application de la Geometrie Imaginaire a quelques Integral as.
Crelle. 1836.
V. Geometrische Untersuchungen zur Theorie der Parallellinien. Ber-
lin, 1840. 61 pages.
VI. Pangeometrie, ou precis de geometrie fondee sur une theorie gene-
rale et rigoureuse des paralleles. Imprimerie de TUniversite. Kazan, 1856.
This, originally published in French, has been translated into Italian by
G. Battaglini : Giornale di Matematiche. Anno V, Settembre e Ottobre,
1867, pp. 273-320. It is also given by Erman, Archiv Russ. XVII, 1858,
pp. '397-456. V has also been translated and published in French ; see
Hoiiel.
.2. Gauss, C. J.
I. Briefwechsel zwischen Gauss und Schumacher. See especially the
letters of 17 May and 12 July, 1831. Bd. 2, pp. 268-271.
**La Geometrie non-Euclidienne ne renferme en elle rien de contra-
dictoire, queique, a premiere vue, beaucoup de ses resultats aient Tair de
paradoxes. Ces contradictions apparents doivent etre regardees comme reflPet
d'une illusion, due a Thabitude que nous avons prise de bonne heure de
considerer la geometrie Euclidienne comme rigoureuse."
Halsted, Bibliography of Hyper-Space and Non- Euclidean Geometry. 263
II. Werke. Bd. IV, p. "215. This reference is to the researches pre-
sented in 1827 to the Society of Gottingen under the title : " Disquisitiones
generales circa superficies curvas."
3. BoLYAi, Wolfgang and Johann.
I. Tentaraen Juventutem studiosara in elementa Matheseos purae, ele-
mentaris ac sublimioris, methodo intuitiva, evidentique huic propria, intro-
ducendi. Tomus Primus, 1832 Secundus, 1833. 8o. Maros-Vasarhelyini.
These two volumes, published by subscription, form the principal work of.
Wolfgang Bolyai. In the first volume, with special title page and number-
ing, appeared the celebrated Appendix of Johann Bolyai,
II. Ap., scientiam spatii absolute veram exhibens : a veritate aut falsitate
Axiomatis XI Euclidei (a priori haud unquam decidenda) independentem.
Auctore Johanne Bolyai de eadem, Geometrarum in Exercitu Caesareo Regio
Austria<50 Castrensium Captaneo. Maros-Vasarhely., 1832. (26 pages of
text). This celebrated Appendix has been translated into French, see Hoiiel,
into Italian, see Battaglini, and into German, see Frischauf.
III. The last work of Wolfgang Bolyai, the only one he composed in
German, is entitled : Kurzer Grundriss eines Versuches, I. die Arithmetik,
durch zweckmassig construirte Begriife, von eingebildeten und unendlich-
kleinen Grossen gereinigt, anschaulich .und logisch-streng darzustellen : II.
In der Geometrie, die Begriffe der geraden Linie, der Ebene, des Winkels
allgemein, der winkellosen Formen, und der Krummen, der verschiedenen
Arten der Gleichheit u. dgl. nicht nur scharf zu bestimmen, sondern auch ihr
Sein in Raume zu beweisen : und da die Frage, ob zwei von der dritten geschnH-
tene Geraden^ wenn die Summa der inneren Winkel nicht zz 2^, sich schneiden
Oder nicht?, niemand auf der Erde ohne ein Axiom (wie Euclid das XI)
aufzustellen, beantworten wird ; die davon unabhangige Geometrie abzuson-
dern, und eine auf die Ja Antwort, andere auf das Nein so zu bauen, dass die
Formeln der letzen auf ein Wink auch in der ersten giiltig seien. Maros-
Vasarhely., 1861. 8o. (88 pages of text). The author mentions Lobat-
chewsky's Geometrische Untersuchungen, Berlin, 1840, and compares it with
the work of his son Johann Bolyai, '' au sujet duquel il dit : ' Quelques
exemplaires de Touvrage publie ici ont ete envoyes a cette epoque a Vienne,
a Berlin, a Gottingen. ... De Goettingen, le geant mathematique, [Gauss]
264 Halsted, Bibliography of Hyper-Space and Non-Euclidean Geometry.
qui du sommet des hauteurs embrasse du iTieme regard les astres at la
profondeur des abimes, a ecrit qu'il etait ravi de voir execute le travail qu'il
avait commence pour le laisser apres lui dans ses papiersJ
J n
4. Jacobi, C. G. J.
I. De binis quibuslibet functionibus homogeneis, &c. Crelle Journ. XII,
1834. 1-^9.
Several papers in the early volumes of Crelle in effect relate to the
transformation of coordinates, and the attraction of spherical shells, &c., in
n-dimensional space, but the treatment is throughout analytical and there is
no especial reference to space of four or more dimensions.
5. Grassmann, H.
I. Die lineale Ausdehnungslehre. Leipzig, 1844. 2d Ed., 1878.
II. Die Ausdehnungslehre. Berlin, 1862.
Grassmann was perhaps the first who developed the theory of extended
manifoldness, as a special case of which appears the theory of space. But
his manifoldness diifers from o\ir space only as being a generalisation of it by
increasing the number of dimensions while preserving relative properties of
position and measure. In a word it is homaloidal Hyper-space, and does not
open so wide and diverse a field as Riemann's profound paper.
6. Cayley, Arthur.
I. Chapters in the Analytical Geometry of (n) Dimensions. Camb.
Math. Journ., IV, 1845. pp. 119-127.
II. Sixth Memoir upon Q.uantics. Phil. Trans., vol. 149.
III. On the Non-Euclidean Geometry. Clebsch, Ann. V, 630-634. 1872.
IV. A Memoir on Abstract Geometry. Phil. Trans., CLX, 51-63. 1870.
V. On the superlines of a quadric surface in five dimensional space.
Quarterly Journ., vol. XII, 176-180. 1871-2.
In his Memoir on the principles of an Abstract m-dimensional Geometry
(IV), Prof. Cayley says: "The science presents itself in two ways, — as a
legitimate extension of the ordinary two- and ^Ar^^-dimensional geometries ;
and as a need in these geometries and in analysis generally. In fact when-
ever we are concerned with quantities connected together in any manner, and
Halsted, Bihlwgraphy of Hyper-Space and Non-Euclidean Geometr-y. 265
which are, or are considered as variable or determinable, then the nature of
the relation between the quantities is frequently rendered more intelligible by
regarding them (if only two or three in number) as the coordinates of a point
in a plane or in space; for more than three quantities there is, from the
greater complexity of the case, the greater need of such a representation ; but
this can only be obtained by means of the notion of a space of the proper
dimensionality ; and to use such representation, we require the geometry of
such apace. An important instance in plane geometry has actually presented
itself in the question of the determination of the number of the curves which
satisfy given conditions : the conditions imply relations between the coeffi-
cients in the equation of the curve; and for the better understanding of these
relations it was expedient to consider the coefficients as the coordinates of a
point in a space of the proper dimensionality."
7. Sylvester, J. J.
I. On certain general Properties of Homogeneous Functions. Cam. and
Dub. M. Journ., Feb. 1851.
II. Partitions of Numbers. (Lectures). London, 1859.
III. Barycentric Projections. Phil. Mag., or Br. Assoc'n.
IV. Inaugural Address to Math. Section, British Association at Exeter,
August, 1869. Nature, vol. I, p. 238. Republished with Notes in " Laws
of Verse." Longmans. 1870.
8. RiEMANN, B.
I. Ueber die Hypothesen welche der Geometrie zu Grunde liegen.
Habilitationsschrift von 10 Juni, 1854. Abhandl. der Konig. Gesellsch. zu
Gottingen. B. XIII. Reprinted in "B. Riemann's G. M. Werke," Leipzig,
1876.
This profound paper is difficult reading. Frischauf has attempted to
make the study of it easier by giving, in his Absolute Geometry, notes and
references at points where Riemann has given results while suppressing
processes.
It has been translated into French by Hoiiel. Annali di Mat., serie II,
tome III, fasc. IV, 309-327. 1870.
■ The position of a point in space being determined by three quantities,
^11 ^11 ^s I to a continuous change of that position corresponds a continuous
266 Halsted, Bibliography of Hyper-Space and Non-Euclidean Geometry.
variation of these three quantities. Then Riemann holds that the measure of
the distance between the point {x^ XzXs) and the next point {Xi + dxi, X2 + dx^,
x^ + ^0-3) is not necessarily the square root of the sura of the squares of the
three diflferentials.
If the sides of a triangle constructed on a given sphere be all of them
increased or diminished in the same proportion, the shape of the triangle will
not remain the same. On the contrary, the figures constructed in a plane
may be magnified or diminished to any extent without alteration of shape.
Riemann found that this property of the plane is equivalent to the two
following axioms: (1) That two geodesic lines which diverge from a point
will never intersect again, or, as Euclid puts it, that two straight lines cannot
enclose a space ; and (2) that two geodesic lines which do not intersect will
make equal angles with every other geodesic line. Deny the first of these
axioms, and you have a manifoldness of positive curvature; deny the second,
and you have one of negative curvature. The plane lies midway between
the two, and its curvature is zero at every point. Thus Riemann found three
diflferent sorts of geometry. Bolyai had only noticed two. Also here for the
first time was brought forward the distinction between '* unbegrenzte " and
'* unendliche " *'Unendliche" is our "infinite." A series is "unbegrenzte"
when, without inversion of the derivation process, one can go on continually.
If one by continued forward application of this process comes back to the
starting point, the series is finite; but if the process can go on continually
without ever coming again to any previous term, the series is infinite. The .
like parts of a circle may serve as an example of a series which though finite
is yet unbegrenzt, for we may pass continually on from one to the next for-
ever. Now we rightly attribute to space this property of being without a
bound, for a limit to it is contradicted by its homogeneousness. But from
this it in no way follows that space is infinite.
9. Salmon, George.
I. Lessons on Modern Higher Algebra. 1866. p. 212, &c.
II. Extension of Chasles' Theory of Characteristics to Surfaces. •
10. Baltzer, R.
I. Elements of Mathematics. Dresden, 1866.
II. Ueber die Hypothesen der Parallelentheorie. C. Journal. Band 83,
s. 372. Berichte der K. s. G. zu Leipzig. T. XX, 96-96. 1868.
Halsted, Bibliography of Hyper-Space and Non-Euclidean Geometry. 267
11. HotJEL, J.
I. £tudes Georaetriques sur la Theorie des Parallels, par Lobatchewsky ;
suivi d'lin extrait de la correspondance de Gauss et de Schumacher. Paris,
1866. 8o. .
II. Essai critique sur les Principes fondamentaux de la Geometrie.
Paris, 1867. 8o.
III. La Science Absolue de TEspace independante de la verite ou de la
faussete de I'Axiome XI d'Euclide (que Ton ne pourra jamais etablir a
priori) ; par Jean Bolyai : precede d'une Notice sur la Vie et les Travaux de
W. et J. Bolyai, par M. Fr. Schmidt. Paris, 1868. 8o.
IV. Sur les hypotheses qiii servent de fondement a la geometrie, memoire
posthume de B. Riemann. Annali di Mat , serie II, tome III, fasc. IV, 309-
327. 1870.
V and VI. Beltrami's " Geometria non-Euclidea" and ** Spazii de cur-
vatura costante," translated into French. Annales Scien. de TEcole Normale
Superieure, tome VI. 1869.
VII. Note sur Timpossibilite de demontrer par une construction plane
le principe de la theorie des paralleles dit Postulatum d'Euclide. Memoires de
la Societe des Sciences de Bordeaux, tome VIII. 1870-72. Paris: J. B.
Balliere.
VIII. Du role de I'experience dans les sciences exactes. Prague, 1875.
Translated into German by F. Miiller. Grunert's Archiv., vol. 59, p. 65.
a
12. Beltrami, E. • .
I. Risoluzione del problema di riportare i punti di una superficie sopra
un piano in modo che le linee geodetiche vengano rappresentate da lines
rette. Annali di Mat , tome VII. 1866.
II. Saggio di Interpretazione della Geometria non-Euclidea. Naples,
1868. Giornale de Matematiche, (G. Battaglini,) Anno VI, pp. 284-312.
III. Teoria fondamentale degli Spazii di Curvatura costante. Annali di
Mat., ser. II, tome II. Milano, 1868.
By the curvature of a system Riemann and Beltrami understand the
relation of the area of an infinitesimal triangle of the system to the corre-
sponding area of a system of constant positive curvature (systeme spherique).
It is in this sense that Beltrami's pseudospherical systems have a constant
negative curvature.
268 Halsted, Bibliography of Hyper-Space and Non-Euclidean Geometry.
This diifers from what Kronecker calls the curvature of a system or^at
the end of his Memoir, the condensation of the system, which instead of the
relation of the areas of infinitesimal triangles, means the relation of the
volumes of infinitesimal tetrahedrons.
IV. Theoreme de Geometrie pseudospherique. Giornale di Mat. This
shows the connection between certain straight lines in the non-Euclidean
plane and the curve whose tangents are of a constant length in the Euclidean
plane.
V. Sur la surface de revolution qui sert de type aux surfaces pseudo-
i^heriques. Giornale di Mat (G. Battaglini), tome X. 1872. This contains
several theorems relative to the surface of revolution having for meridian the
c^irve whose tangents are of a constant length.
13. Battaglini, G.
I. Sulla Geometria Immaginaria di Lobatchewsky. Giornale di Mat.,
Anno V, pp. 217-231. 1867. In this the author reaches by a different
method most of Lobatchewsky's results.
II. Pangeometria o sunto di geometria fondata sopra una teoria generate
6: rigorosa delle parallele, per N. Lobatchewsky, (versione del Francese).
Giornale di Mat., Anno V, pp. 273-320.
III. Sulla scienza dello spazio assolutamente vera, ed independents dalla
verita o dalla falsita dell' assioma XI di Euclide : per Giovanni Bolyai, (ver-
sione dal latino). Giornale di Mat, Anno VI, pp. 97-115. 1868.
14. Helmholtz, H.
I. Ueber die Thatsachen die der Geometrie zum Grunde liegen. Nach-
richten, Giittingen, Juni 3, 1868.
II. Sur les faits qui servent de base a la Geometrie. Memoires de la
Soc. des Sciences de Bordeaux. 1868.
III. The Origin and meaning of Geometrical Axioms. Part I, Mind, No.
III. July, 1876. Some of this article had been previously given in the
Academy, Feb. 12, 1870, vol. I, p. 128 Replied to by Jevons; Nature, vol.
ly, p. 481. Jevons' ideas developed by J. L. Tupper; Nature, vol. V, p. 202.
Replied to by Helmholtz ; Academy, vol. Ill, p. 52. Part II, Mind. April,
1878.
Halsted, Bibliography of Hyper-Space and Non-Euclidean Geometry, 269
15. POTOCKI, S.
Notice historique sur la vie et les travaux de N. I. Lobatchewsky. Bul-
letino di Bibliographia du Prince Boncompagni, tome II, 223. May, 1869.
Translated from the discourse of Janichefsky, who was editing a new edition
of Lobatchewsky 's works.
16. Darboux, G.
I. Comptes Rendus de 1' Acad. Aug., 1869.
II. Sur les equations aux derivees partielles du second ordre. Comp.
R. LXX. 1870. I, 673 ; II, 746,
17. Kronecker, L.
I. Ueber Systeme von Functionen mehrer Variabeln. Monatsbcricht
der Kgl. Akademie zu Berlin. Part I, Marz, 1869. Part II, August, 1869.
The generalized spaces treated here are mostly supposed homaloidal. The
author mentions the power given him by considerations of geometry of
position in overcoming algebraical difficulties.
18. Christoffel, E. B.
I. Allgem. Theorie d. geodat. Dreiecke. Berlin, 1869.
II. Ueber die Transformation der homogenen Differentialausdriicke 2^®"
Grades. Borchardt's Journal, LXX, 46-70. 1870.
III. Ueber ein betreifendes Theorem. Borchardt's Journal, LXX, 241-
245. 1870. II and III treat of n dimensions.
19. Clifford, W. K.
I. On Probability. Educational Times.
II.« Lecture on " the Postulates of the Science of Space." The extent of
space may be a finite number of cubic miles. He says, " In fact, I do not
mind confessing that I personally have often found relief from the dreary
infinities of homaloidal space in the consoling hope that, after all, this other
may be the true state of things."
III. Preliminary sketch of Biquaternions. Proceedings of L. Math.
Soc, IV, 381-395. The author shows that the symbols have a more general
interpretation in the geometry of three dimensions which Klein calls the
• elliptic in distinction from the parabolic or Euclidean geometry.
68
270 Halsted, Bibliography of Hyper-Space and Non-Euclidean Geometry.
20. LiPSCHiTz, R.
I. Untersuchungen in Betreff die ganzen horaogenen Functionen von n
Differentialen. Borchardt's Journal, Bde. LXX, 3, pp. 71-102. LXXII, 3,
pp. 1-56. Analysed in the proceedings of the Berlin Academy, Jan., 1869,
pp. 44-53. An analysis by the author is given in the Bulletin des Sciences
Mathematiques, tome IV; I, pp. 97-110; II, pp. 142-157. Paris, 1873.
The author demonstrates that the general form of the linear element of a
system of three dimensions can be referred back to the form given by Rie-
mann for a system of constant curvature, when a certain condition necessary
and sufficient, (that the measure of the constant curvature shall be equal to
a given function) is satisfied.
II. Entwickelung einiger Eigenschaften der quadratischen Formen von
n Diiferentialen. Borchardt's Journal, LXXI, 274-287, 288-295. Bulletin
des Sciences Math , IV, 297-307 ; V, 308-314 Paris, 1873.
III. Untersuchung eines Problems der Variationsrechnung. Borchardt's
Journal, Bd. LXXIV, pp. llC-149, 150-171. Bulletin, tome IV, 212-224,
297-320. .
IV. Extension of the Planet-problem to a space of n dimensions and of
constant integral curvature. Translated by A. Cayley. Quar. Jour. Math.
XII, 349-370. 1871.
21. Genocchi, a.
Dei primi principii della mecanica e della geometria in relazione al pos-
tulate d'Euclide. Firenze, 1869. Accademia da XL in Modena, serie III,
tomo II, parte I. This memoir connects the theory of parallels and parallel
forces with mechanical laws and considerations.
22. NOTHER, M.
Zur Theorie der algebraischen Functionen mehrerer complexer Variabeln.
Gottingen, Nachrichten. 1869.
23. Betti, E.
Sopra gli spazi di un numero qualunque di dimensioni. Annali di Mat.,
2 serie, IV, pp. 140-158. 1870. Contains analytical treatment of the prop-
erties and relations of spaces of equal or different dimensions.
Halsted, Bibliography of Hyper- Space and Non-Euclidean Geometry. 271
24. DE Tilly, M.
I. £tudes de mechanique abstraite. Memoires couronnes de rAcademie
royale Belgique, tome XXI.
II. Report on a letter from Genocchi to Qiietelet. Bulletin de Belg. (2) •
XXXVI, 124-139.
25. Becker, J. K.
I. Abhandlungen aus dem Grenzocebiete der Mathematik und Philoso-
phie. Zurich, 1870. (62 pages).
II. Ueber die neuesten Untersuchungen in Betreff unserer Anschauungen
vom Raume. Sehlomilch Zeitschrift, XVII, 314-332. 1872. For recension
of I see XV, 93.
III. Die Elemente der Geomet.rie auf neuer Grundlage. Berlin, 1877.
(300 pages). This contains a systematic statement of the ground-principles
of the plane and of space which appear in the properties of the simplest
figures. The moving idea is that all the properties of figures are grounded
in the nature of space itself.
26. SCHLAEFLI, L.
I. Nota alia memoria del Sig. Beltrami sugli spazie dela curvatura cos-
tante. Annali di Mat , 2d serie, t V, 178-193. 1870.
II. Beltrami. Osservazione sulla precedente Memoria del Sig. Prof.
Schlafli. Brioschi, Ann. V, 194-198. A theorem of Beltrami leads to tlie
problem : To distinguish all spaces of n-dimensions in which any geodetic
line is represented by a system of n — 1 linear equations. Schlafli shows that
only spaces of constant curvature fulfil this condition.
27. Beez, R.
I. Ueber conforme Abbildung von Mannigfaltigkeiten hoherer Ordnung.
Sehlomilch Zeits., XX, 253-270.
II. Zur Theorie des Krummungsmasses von Mannigfaltigkeiten hoherer
Ordnung. Sehlomilch Z., XX, 423-444. Fortsetzung, XXI, 373-401. The
author shows that Kronecker's generalized expression for the measure of
curvature for Hyper-space cannot, as in tridimensional space, be represented
by the coefficients of the expression for the linear element. In reference to
this, Lipschitz (Beitrag zur Theorie der Krummung. Borchardt's Journ.,
272 Halsted, Bibliography of Hyper-Space and Non-Euclidean Geometry.
LXXXI, 239, Note) remarks, that to make the representation possible, one
has only to take in addition ** die Difterentialquotienten jener CoeflBcienten
nach den Variabeln."
28. ROSANES, J.
Ueber die neuesten Untersuchungen in Betreif unser Anschauung vom
Raume. Breslau, 1871. 8o. This is an elementary exposition of the ideas
contained in Riemanii's celebrated paper.
29. Flye, St. Marie.
I. Sur le postulatum d'Euclide. L'Institut, I, sect. XXXVIII, 53-64.
1870. That the postulatum of Euclid cannot be proved except by assuming
another of like value.
II. £tudes analytiques sur la theorie des paralleles. Paris, 1871. 8o.
Treats of a system of coordinates whose axis of a* is a circle with infinite
radius.
30. Lie, Sophus.
I. Ueber diejenige Theorie eines Raumes mit beliebig vielen Dimensi-
onen, die der Kriimmungs-Theorie des gewohnlichen Raumes eptspricht.
Gottingen, Nachrichten. May, 1871. In this many geometrical theorems
are extended to a space of any number of dimensions.
II. Zur Theorie eines Raumes von n Dimensionen. Gottingen, Nach-
richten. Nov., 1871. 535-557. The sphere of n dimensions is used as
element of a space oi n + 1 dimensions.
31. Klein, Felix.
I. Ueber die sogenannte Nicht-Euklidische Geometrie. Gottingen, Nach-
richten. August, 1871. Math. Ann., IV, 573-625; VI, 112-145. The
projective geometry is proved to be independent of the theorem of parallels.
See Jahrbuch uber die Fortschritte der Math., 1873.
II. Ueber neuere geometrische Forschungen. Erlangen, 1872.
32. Saleta, F.
Expose sommaire de Tidee d'espace au point de vue positif. Paris, 1872.
(32 pages.) The author considers the axioms and postulates of geometry as
definitions of the kind of space treated.
Halsted, Bibliography of Hyper-Space and Non- Euclidean Geometry. 273
33. KoNiG, J.
Ueber eine reale Abbildung der Nicht-Euklidischen Geometrie. Got-
tingen, Nachrichten. March, 1872. (7 pages). A study of the relations
which exist between the non-Euclidean geometry and the geometry of com-
plexes.
34. Jordan, Camillb.
I. Essai sur la Geometrie a n Dimensions. Comptes Rendus, LXXV,
1614-1617. 1872. Bulletin de la Soc. Math., tome III, pp. 104, &c., IV,
p. 92.
II. Sur la theorie des courbes dans I'espace a n dimensions. Comptes
Rendus, LXXIX, p. 795. 1874.
III. Generalisation du theoreme d'Euler sur la courbure des surfaces
dans I'espace a m + ^ dimensions. Compt. Rendus, LXXIX, p. 909.
35. Frischauf, J.
I. Absolute Geometrie, nach J. Bolyai. Leipzig, 1872. 8o. XII-96 pp. *
II. Elemente der Absoluten Geometrie. Leipzig, 1876. VI-142 pages.
36. KOBER, J.
On infinity and the new geometry. Zeits. fur Math. Unterricht. 1872.
37. Hoffmann, J. C. V.
Resultate der Nicht-Euklidischen oder Pangeometrie. Zeits. fiir Math.
Unterricht, IV, 416-417.
38. Freye, G.
TJeber ein geometrische Darstellung der imaginairen Gebilde in der
Ebene. Jena. Neuenhahm.
39. Cassani, p.
Intorno alle ipotesi fondamentali della geometria. Battaglini, G. XI,
333-349.
40. Frahm, W.
Habilitationsschrift. Tiibingen, 1873.
41. LlNDEMANN, F.
Ueber unendlich kleine Bewegungen starrer Korper bei allgemeiner pro-
jectivischer Massbestimmung. Erlang., Ber., 1873, 28 Juli. Clebsch, Ann.
VII, 56-144.
69
274 Halsted, Bibliography of Hyper- Space and Non- Euclidean Geometry.
42. d'Ovidio, E.
Studio sulla geometria projettiva. Brioschi, Ann. (2) VI, 72-101.
43. Stahl, H.
Ueber die Massfunctionen der analytischen Geometrie. Berlin, 1873.
44. SCHERING, E.
Linien, Flachen und hcihere Gebilde im mehrfach ausgedehnten Gauss'
schen und Riemann' schen Raume. Gottingen, Nachrichten, 13-21; 149-159.
1873. For a notice of the last seven by Klein, see Jahrbuch iiber die Fort-
schritte der Math. Berlin, 1875.
45. Spitz, C.
Die ersten Satze vom Dreiecke und den Parallelen. Nach Bolyai's
Grundsatzer. Leipzig. For notice see Grunert's Archiv., LVII, Litber,
CCXXVI, 10.
46. Halphen, G.
Recherches de geometrie a n dimensions. Bull. Soc. Math., F. II, 34-52.
On the projective properties of structures in Hyper-space.
47. ESCH ERICH.
Die Geometrie auf den Flachen constanter negativer Krummung. Kais.
Akad. Bd. LXIX. .
48. Spottiswoode, W.
I. Sur la representation des figures de geometrie a n dimensions par les
figures correlatives de geometrie ordinaire. Comptes Rendus, LXXI, 875-877.
II. Nouveaux exemples de la representation, par des figures de geometrie,
des conceptions analytiques de geometrie a n dimensions. C. R., LXXXI,
961-963. Geometrical interpretation of analytical space of more than three
dimensions.
49. Lewes, G. H.
Imaginary Geometry and the truth of Axioms. Problems of Life and
Mind, 1st series, vol. II. London, 1875.
«
50. Funcke.
Grundlagen der Raumwissenschaft. Hannover, 1875. Proposes a case
in which he holds it would be imperatively necessary to suppose a fourth
dimension.
Halsted, Bibliography of Ht/per- Space and Non- Euclidean Geometry. 275
51. ZOLLNER^ J. C. F.
I. Principien einer Elektrodynamischen Theorie der Materia. Leipzig,
1876. See Review by Prof. Carl Sturnpf, Phil. Monatshefte, B. XIV, 13-30.
II, Wissenschaftliche Abhandlungen. Leipzig, 1878. See Review by
P. G. Tait, Nature, March 28, 1878, pp. 420-422.
52. Frank, A. von.
Der Korperinhalt des senkrechten Cylinders und Kegels in der absoluten
Geometrie. Grunert's Archiv., vol. 59, p. 76.
53. GtJNTHER, SlEGMUND.
^^ •
Ziele und Resultate der neuern Math. Histor. Forschung. Erlangen,
1876. Also Reviews, Grunert's Archiv., Theil 60.
54. Rethy.
Die Fundamental gleichungen der nicht-Euklidischen Trigonometrie auf
elementarem Wege abgeleitet. Grunert's Archiv., LVIII, 416.
55. Frankland, W.
On the simplest continuous manifoldness of two dimensions and of finite
extent. Nature, vol. 15, No. 389, April 12, 1877, pp. 515-517. This article
called forth an objection from C. J. Monroe in Nature, vol. 15, No. 391, April
26, 1877, p. 547, where he claims that it necessitates that a perpendicular
change sign without passing through Infinity or vanishing. Of this objection
Prof. Newcomb says: " I cannot see even what it consists in. The first ele-
ments of complex functions imply that a line can change direction without
passing through Infinity or zero."
56. Erdmann, Benno.
Die Axiome der Geometrie. (Untersuchung der Riemann-Helmholtz
Raum theorie). Leipzig, 1877.
57. Mehler.
^ Ueber die Benutzung einer vierfachen Manningfaltigkeit zur Ableitung
orthogonaler Flachensysteme. Borchardt's Journ., Band 84, pp. 219-230.
December, 1877.
276 Halsted, Bibliography of Hyper-Space and Non-Euclidean Geometry.
58. Cantor, G.
Ein Beitrag zur Mannigfaltigkeitslebre. Borchardt's Journ., Band 84,
pp. 242-258. December, 1877.
59. Newcomb, Simon.
I. Elementary theorems relating to the geometry of a space of three
dimensions and of uniform positive curvature in the fourth dimension. Bor-
chardt's Journ., Band 83, pp. 293-299. 1877.
This article, founded on the ideas of Riemann, considers the subject from
the standpoint of elementary geometry. *' It may also be remarked that
there is nothing within our experience which will justify a denial of the
possibility that the s{)ace in which we find ourselves may be curved in the
manner here supposed."
II. Xote on a class of Transformations which Surfaces may undergo in
Space of more than three dimensions. American Journal of Math., I,
pp. 1-4. 1878.
** If a fourth dimension were added to space, a closed material surface
(or shell) could be turned inside out by simple flexure; without either
stretching or tearing."
60. Tannery, Paul.
I. La Geometric Imaginaire et la Notion d'Espace. Revue Philoso-
phique, Nov., 1876, 433-451. II, No. 6, pp. 553-575, Juin, 1877.
61. LtlROTH, J.
Ueber Bertrand's Beweis des Parallelenaxioms. Schlomilch Zeitschrift,
XXI,' pp. 294-297. 1876. Pointing out the failure of that attempted demon-
stration.
62. Weissenborn, H.
Ueber die neueren Ansichten vom Raum und von den geometrischen
Axiomen. Vierteljahrschrift fiir Wissenschaftliche Philosophic, II, Zweites
Heft, Erstes Artikel, 222-239. 1878.
SOME REMARKS ON A PASSAGE IN PROFESSOR SYLVESTER'S
PAPER AS TO THE ATOMIC THEORY.
Contained in a Letter addressed to the Editors by Peofessor J. W. Mallet,
of the University of Virginia.
In the paper, published in your Journal, by Professor Sylvester on "An
application of the new atomic theory to the graphical represeiltation of the
invariants and covariants of binary quantics," there .occurs* a suggestion
which, although parenthetic only as to the main subject of the paper, seems
to deserve careful consideration and more prominence than its able author
has deemed it worthy of. Professor Sylvester suggests, namely, as a modifi-
cation of the atomic theory of chemists in its present form, that " leaving
undisturbed the univalent atoms, every other n-valent atom be regarded as
constituted of an n-ad of trivalent atomicules arranged along the apices of a
polygon of n sides." Thus, using small letters to stand for the proposed
"atomicules," an atom of univalent hydrogen would be graphically repre-
sented by h — , and a molecule of the same element by h — h , as at present ■
' an atom of bivalent oxygen by — o^o — , and a diatomic molecule of the
same by | | ; an atom of trivalent nitrogen by \ / , and a diatomic
i
molecule by J ? ] ; an atom of quadrivalent carbon by | j , and a diatomic
molecule by rt ^j ; &c.
Jc+i
Two important advantages of this supposition are pointed out by Pro-
fessor Sylvester. First, that it furnishes a conceivable explanation of the
existence, in the isolated state, of single atoms of mercury, cadmium, &c.,
• Pftge 78.
278 • Mallet, Some Bemarks on the Atomic Theory.
which may be represented as composed of two trivalent atomicules united
by all three bonds, thus ...
hg EE hg ,
while in their ordinary compounds the same atomicules would have but two
bonds hxter se^ thus
— hg = hg — .
Secondly. The explanation oifered by Frankland of the variability of the
valence of an atom (the weakest point in the theory of atomicity), viz., that
" one or more pairs of bonds belonging to the atom of an element can unite
and having saturated each other become, as it were, latent," has always
seemed to me to carry with it absolutely no physical meaning — the reaction
of a single and indivisible centre of chemical force upon itself may fairly be
called " unthinkable." If, however, we admit Professor Sylvester's concep-
tion of an atom as made up of chemically inseparable but yet discrete
atomicules, susceptible of force relation among themselves and to a variable
extent, Frankland's idea assumes intelligible form.
To these results following from the suggested modification, I venture to
add one or two others.
P. A graphic representation is aiforded of the difference between the
old " equivalent weights " and the atomic weights of the elements ; a differ-
ence which for many years formed the chief subject of controversy as to the
atomic theory, and the chief stumbling block in the way of further progress
in its application. Thus, if we take the old definition of the equivalent
weight of an element, that it is *'the smallest quantity of it which unites
with 1 part of hydrogen," we have for oxygen 8, for nitrogen 4§ , for carbon 3,
&c. ; while other considerations, especially that of the fractional replacement
of the hydrogen in such compounds as have yielded these figures, oblige us
to assign the atomic weight 16 to oxygen, 14 to nitrogen, 12 to carbon, &c.
If the following graphs be taken to represent the compounds in question —
water, ammonia and marsh gas,
h
\
h
/
h h
\ /
1 u
\/
n
c c
li
/ \
h h
we see that
the weight of the atom of oxygen being 16, of nitrogen 14, of carbon 12,
the weight of the atomicule of oxygen is 8, of nitrogen 4§, of carbon 3, &c.
Mallet, Some Remarks on the Atomic Theory. 279
2°. I venture to add still further a slight modification to the suggestion
of Professor Sylvester.
So long as we regard atoms as the ultimate units of matter, we can, in
the present state of our knowledge, assume nothing as to their size or shape;*
indeed the greater number of chemists and physicists are perhaps rather
inclined to look upon them as mere "centres of force." If, however, we
consider an atom as made up of non-coincident atomicules, though these
latter be but points, we may be quite ignorant of the size and shape of such
a system of points, but both size and shape must clearly be predicable of it
taken as a whole. And, in like manner (whether we adopt this idea of
atomicules or not), molecules, made up of non-coincident atoms, must be
possessed both of size and shape, whether these be by us determinable or not,
and w^hether they be invariable or' subject to change. Let it be assumed that
an atom, when consisting of two or more atomicules, constitutes a rigid sys-
tem, of invariable size and shape, the atomicules preserving permanently the
same positions in relation to each other. But in a molecule made up of
atoms, let it be assumed that the relative position of the atoms admits of
change, and hence that in consequence of chemical combination, decom-
position, substitution, &c., distortion of the molecules occurs, on which dis-
tortion may depend, in part at least, the changed properties of the masses
made up of such molecules in aggregate.
If this idea be admitted I believe that an explanation may be found of
the heat relations between the molecules of the two allotropic forms of
oxygen, which possess special importance from their having the simplest
connection with the atomic theory of all known cases of allotropism.
It is now generally agreed that the molecule of ordinary oxygen consists
of two atoms, and that of ozone of three. The conversion of two molecules
of the latter into three • of the former is attended with evolution of heat, to
an extent estimated by Berthelot from his calorimetric experiments at 59.200
heat units ; and, conversely, an equivalent amount of extraneous energy must
be exerted in the production of two ozone molecules from three •of ordinary
oxygen. If the usual graphs for these two bodies be employed, there is no
reason apparent for the thermic relation in question ; the number of bonds
* This assertion has nothing to do with the calculations of Sir William Thomson and others as to the ** size
of atoms," since these calculations are only concerned with the dimensions of the sphere of mutual interaction of
mechanical atoms or molecules, and do not at all apply to actual occupancy of space by the atoms themselves.
280 Mallet, Some JReniarks on the Atomic Theory.
for each atom and the average distance between it and the other atoms with
which it is connected is just the same in
q — o
0=0 and \ /
(ordinary oxygen.) (ozone.)
But if Professor Sylvester's graphs be substituted, namely
0=0
= / \
1 I and o o
0=6 \ //
o — o
(ordinary oxygen.) (ozone.)
it is at once suggested to the eye* that the average distance between the
atoms, as also between the atomicules, is increased in passing from the
ordinar}- form of oxygen to ozone ; or, in other words, that the production of
ozone from oxygen gas is an act of partial chemical decomposition, not result-
ing in the detachment of the constituent atoms from each other, on the con-
trary leaving them in a state of definite chemical combination, but removed
further apart than they were in the more stable form of the element —
hence, as in complete, so in this which I have called partial decompositioa,
extraneous energy is necessary; while the reverse change, from ozone back
to ordinary oxygen, is an act of more intimate chemical union, resulting in
closer approximation of the constituent parts of the molecule — and hence
attended with evolution of energy, as in the form of heat.
In conclusion, I would invite the attention of mathematicians to the
interesting field for their examination which is being rapidly opened up
by the study of the thermic changes accompanying chemical action. The
researches of Thomson and Berthelot in particular, are fast accumulating a
large mass of numerical data as to heat evolved or disappearing in connec-
tion with chemical combination and decomposition, and the highest interest
attaches to a proper discussion of such results in the light of the atomic
theory. In too many reviews of the facts already before us there has been
gross neglect even of the thermic relations of changes of physical state accom-
panying chemical action, but, even where these and other collateral phe-
* It is no valid objection to this to say that the arrangement of the four atomicules in the one case in the
form of a square, and of the six in the other case us a regular hexagon, is merely imaginary, for the same
result as to average distance will follow from any arrangement which is symmetricnl (and with any aasumption
as to relative distances between atomicules and atoms), and dealing, as we are here, solely with timilar atoms and
atomicules we can scarcely avoid the belief that their arrangement, whatever it may be, is symmetrical.
Mallet, Some Remarks on the Atomic Tlieory. 281
nomena have been duly allowed for, there has been so far no proper consid-
eration of the force involved in the union or separation of similar as well as
dissimilar atoms. Thus we find the formation of hydrochloric acid from its
elements represented as
H + CI = HCl = 22.000 heat units,
whereas we really have
H2 + CI2 = 2HC1 = 22.000 h. u.
the + 22.000 h. u. representing the algebraic sum of the thermal changes
involved in the decomposition of a molecule of hydrogen (separation of its
two atoms), the similar decomposition of a molecule of chlorine, and the
formation of two molecules of hydrochloric acid. A mathematical discussion
of the data already on hand might possibly suggest the means of so com-
bining our calorimetric experiments as to reduce the number of unknown
quantities in our equations, and lead ultimately to a clear and connected
view of the force concerned in the various chemical changes which admit of
being accurately examined.
Unit, of Virginia, July iB^ 1878.
71
NOTES.
I.
Historical Data concerning the Discovery of the Law of Valence.
•
At page 89 of the Journal, reference was made to an editorial notice in
Nature (March 14, 1875.) The date should have been February 14, and the
article is of too great historical interest in connection with the Chemico-
Algebraical Theory to be exposed to the chances of being overlooked. It
reads as follows :
" In his interesting communication on the analogy between chemistry
and algebra in our last number, Professor Sylvester attributes the conception
of valence or atomicity to Kekule. No doubt the theory in its present devel-
oped form owes much both to Kekule and Cannizaro ; indeed, until the latter
chemist had placed the atomic weights of the metallic elements upon a con-
sistent basis, the satisfactory development of the doctrine was impossible.
The first conception of the theory, however, belongs to Frankland, who first
announced it in his paper on Organo-metallic Bodies, read before the Royal
Society on June 17th, 1852. After referring to the habits of combination of
nitrogen, phosphorus, antimony and arsenic, he says: 'it is sufficiently evi-
dent, from the examples just given, that such a tendency or law prevails, and
that, no matter what the character of the uniting atoms may be, the comhining
power of the attracting element^ if I may be allowed the term, is always satisfied
by the same number of these atoms' He then proceeds to illustrate this law by
the organo-compounds of arsenic, zinc, antimony, tin, and mercury. In con-
junction with Kolbe, Frankland was also the first to apply this law to the
organic compounds of carbon ; their paper on this subject, bearing date
December, 1856, having appeared in Liebig's Annalen in March, 1857, whilst
Kekule's first memoir, in which he mentions the tetrad functions of carbon,
is dated August 15th, 1857, and was not published until November 30th in
the same year. Kekule's celebrated paper, however, in which this applica-
tion of the theory of atomicity to carbon was developed, is dated March 16th,
1858, and was published on May 19th, 1858. On the other hand, the "chemi-
cographs" or graphic formulae, which Professor Sylvester has so successfully
applied to algebra, were the invention of Crum Brown, although Frankland
has used them to a much greater extent than any other chemist."
282
Notes.
283
II.
On the Mechanical Description of the Cartesian.
By J. Hammond, Bath^ England.
Suppose two thin circular discs -4, B rigidly attached to
each other and capable of revolving round a pin through their
common centre; [or perhaps it would be better to make the
pin and discs revolve together.] Suppose also that fine strings,
ACP^ BJDP, are wrapped round the discs and either passed
through small rings at C, 2>, or twisted once round small pins
there, and then knotted together at P, where a tracing point
is attached.
Then it is evident that (if CP zz r , DP = r^) dr: dr' = ratio of the radii
of A and -B, and that P traces out a Cartesian, with two of its foci at C and D:
When A =: B and one of the strings is wound on to the circle while the other
is wound oflF, the locus of P is of course an ellipse, but when both are wound
off together the locus is an hyperbola.
A simple construction for the tangent to the Cartesian at P is obtained
by constructing a parallelogram of velocities, whose diagonal will be the tan-
gent, the sides being measured along PC, PD proportional to the resolved
velocities of P in those directions, i. e. to the radii of A and B.
III.
A New Solution of Biquadratic Equations.
By T. S. E. Dixon, Chicago, Ills.
Reducing the given equation to the general form, y* + 2py^ + Sqy = r,
let y = ^ + \ — \p + ^ + — - ) and make the substitution. It will result in
the cubic, ^^ + px^ + j ~ '^ = 2^ ? from which the values of x and conse-
quently of y are readily obtained. Let or = V n , and the four values of y are
p + n
^-7i)
y = — Vn + V— (i)+w — -L),
n
V n*
V n^
284 Notes.
If now we substitute for n its value derived from the cubic equation, we
have a compact available formula for the expression of all the roots, free
from the cumbersome — ^^^ and ^z- of the old formulae.
2 2
Moreover, it is necessary to obtain only one value of x in the cubic equation.
The equation ^ — 100/ + 480^ m 576, where n equals 9, 16 and 25, is a
good illustration.
It is thus made clearly manifest that the roots of a biquadratic equation,
wanting its second term, may be expressed in the generjil form yz=. s/ A + s/ B .
This is a general expression for all four roots, since each radical has both a
positive and negative root. Thus
y — sf~A + V^ , y — — '^'^ + '^^ ,
y= ^f~A — s/~B, y — — sf'A — sflR.
B is also a function of V ^ and the expression may be wTitten
y=^/'A + ^lF^/'A .
This leads to the suggestion that, if the roots of an equation of the sixth
degree are capable of algebraic expression, they may possibly assume the
form, y =z V^ + \^ ^ , the two roots of the first and the three roots of the
second radical affording the six necessary variations. Possibly there may be
two general forms, one in which ^ is a function of\/B, and the other in
which ^ is a function of V ^ .
The solution of the following equation would, however, indicate still
another form. If x^ + px* + 2^ + ^^^ + sx = i^a "j* T _ ^ ^ ^j^^j^
p — 4r
~ 4 "^^V 4 >' ^6" 6V7"-F4it>'
3
■ ^— g +Vg^ + 4« lf-q + ^f + 4u y , fp ^ 2s — p q \»
+ 4 ^^ 4 ) + V6 +6V7+S>' ■
Thus, if a^ — Sjc* + GOj-^ — lOS.r'' — 6.r = — 884, x = — 2.
By making one or more of the coefficients p, q, r, s and « equal to zero,
a number of solutions of particular cases in equations of the fifth and sixth
degrees are obtained.
Notes. 286
IV.
On a Short Process for Solving the Irrediunble Case of Cardan's Method.
By Otis H. Kendall, Assistant Professor in the University of Pennsylvania.
The equation having three commensurable roots, a, 6, Cj is
a;^ — (a + 6 + ^) •^ + {ab -\- ac + be) x — abc = .
Reducing the roots of this equation by — (a + ^ + ^) > we have
if^ — -^ («^ + ft^ + ^ — o^ — (^ — bc)y
_ ^ (2a« + 26'^ + 2c« — 3a'6 — 3aV — 3flft^ — ^a(? — Zb^c — ^b(? + Viabc) = 0.
This being of the form y^ + ^y + w = , we have, substituting in Cardan's
formula and reducing.
=4<n
27 . 3
y=~\ — ~n + ^{a — b){a — c){b-c)^^-3
l^
1 ; 27 3
Since « is a binomial imaginary, its cube root will be of tbe form
a + V — 13, and \ ^^will be rational. Hence
(1) a (a- — 3/i^) = -1 12a' + 2¥ + 2c» — 3a*& — Sa'e — 3ab^ - Sac'
— 3ft^c — 36c* + 12abc) ,
(2) V—iJ (3a^ — 13) = -^ (a — J) (a — c) (J — c) V— 3 = (r).
Since a must be rational, V — ^ must be of the first degree with reference to
a, J, c, and the only factors of (r) of that degree are of the form ^ — m. ,
V
b — c
where p is some integer : /> must be 2, for substituting V — ^ = V — 3
V
in (2) and reducing, we have
a^ = -V (« — ^* — ao + be) + r- — ,
2 - i^
which will not give a rational value to a unless /> m 2.
Let us assume then V — ^ = — - — V — 3, substitute in (2) and reduce.
We find a zz db (« — ~~w~) ' ^^^ ^y substitution in (1), a = a — "T ,
and similarly for the other factors of (r). Hence
72
286 Notes.
_ (c — ^i + ^^ V^=^3) ; similarly
''=3('^--2 2-^-^)'3 C*--^^ ^V^3),and
3. V 2 2 / '
^ = — (2a — 6 — '') ' T (^* — ''^ — ^) ' ^^^ "q" (^^ — '^ — J) ; .r = a, J, and c.
The three values of u are connected, as they should be, by the ratios
I , . . . ^.^ 1
1 : ^ (- 1 + V - 3) : ^ (- 1 - V - 3) .
If two of the roots of the equation are equal, as h and c, then (r) = and
o" ^ ^^ (^ — ^)^ ' ^^^ ^^ *^^ ^^^ imaginary, as ft and c^ (= y db 5 V — 3) ,
(r) becomes rational and ^ ^ + W = (^ — 7 — 36)% showing why, in
these two cases. Cardan's formula gives a rational result.
Numerical Equations.
1. o:^ — 9a:2 + 14r + 24 = 0. ^
Reduce the roots by 3; then ^ — 13^ + 12 = 0, and by Cardan's formula,
9 \ ^3^ 3^^ 3^ 3
(1) a(a^-3/3)=-6, (2) V-/3(3a^-/3)=^\/-i;
/ — s *'~T 5 / r , 7 \ r r, 1 25 ,49 ,,
V-^=X-3 , _V— 3 , and-_^,-_;/3 = _, _,and j^;then
by (2), a = ±2, ±|-, and±l; and by (1), a = — 2, |-, andl.
...,=_^+Vr|+(_,_vzj)^=_4,,=|+|V^-+3
6, and 4.
2. .r» — 16x* + 7ar — 90 = .
1 ft ^7 110'
Reduce the roots by — ; then f ~ y — — -- = 0, and by Cardan's formula,
1 ^\~. — _^ . — ^ . 1 «
y=— ^55 + 126V — 3+ |-^55 — 126V — 3.
Notes. 28'
126 , — R 3
Since - =- V — ^ ^^ 2 iP' — ^)(^ — <')(<'' — <') '^ — 3> t^e factors are probably
3 , — ^ , . 7
±2V — 3,±-s-^ — 3 and ± -^V — 3. Hence
(1) a (a^ — 3/3) = 55 , (2)' V'^TJ (3tt^ — /3) = 126 V ^^3 ;
V^& = 2 V^rg ; 3a* =75, a = ±6; a = — 5;
363 .11 11
V_/? = -|-V-3; 3a* = ^, a = ±^; a= ^;
1 1
10
y = ^ (- 5 + 2V- 3) + -t- (- 5 - 2 V- 3) = - y ,
or = 2, 9, and 5.
«
V.
^w Extension of Taylor^ 8 Theorem.
By J. C. Glashan, Ottawa^ Canada.
.•./(x + «) = (l-j^^A)"/(x). I.
(by I,) =j'rfa{l-J^"^*rf(a+6)A|"/(x). II.
Expanding by I, but modifying each remainder by II before proceeding
to obtain the term next following, we get
f{x + a+b + c + e + &c.) =f{x + b + c + e + &c.)+ C daf{x-\-c + e + &c.)
+ CdaC^ d{a + b)f'' {x + e + &Q.)
+ f'daC'^''d{a + b) f^*'^'*; (a + ft + c)/* (^ + «&c.) + &c., III.
which i/thee^Unsion proposed.
288 Notes.
Writing "C for a function of «„, «-,, «2, . . . a,„_,, and "-"u."C? for the
result of substituting, in "'C «, + fl-a + «3 + • • • «.-,. for a,„ a,_, + , for a,,
«»-m + 2 for Oj , &c., that is, if '"C — />' (oo , Oi , 02 .••••««- 1) , then
"-"q.^C^ JF'(r/, + «, + «.,+ ... ff„_„, «„_„ + ,, a„_„ + 2, . . . a,_,) ,
and determining these C'-functions by
"0= oj + - a:r' ("-'(.) .'0 + !L(_p^l) a--^ (»-*« .='0 + .... + ^ a,CG..-'C),
III may be put under the form
/ W =/ (-c — ««) + J ,/' (■'• — «u - «.) + p ^ /" (-r — «o — a, — a*) +
+ -"^ ./" (^- — K^r) + &c. IV.
Writing a for ^/y, then if ^i m <?2 =i <?.=:... = ^n = ^? this becomes the
Series of Abel given by M. J. Bertrand in his Traitede Calcul differ entielj p. 324.
f{x)=f{x- a) + ^f{x-a-h) +"S^±P f{x-{a + 2b)] +. ...
- — ,-- - / {•^•— (« + «*)} + &c.
If J = this becomes Taylor's Theorem, or the simple expansion of I as evi-
dently it should.
Using a, ft, c, rf, f?, &c., instead of (i„y «„ a^, &c., we get for the first five
coefficients of IV,
^C=d'-\- 2ab ,
3C= a" + 3d' {b+ c) -]-Sa {b' + 2bc) ,
*C =a* + 4o'' {b + <• + f/) + 6a^ { {b + f)'' + 2 (ft + ^) d} + 4« {ft* + 3ft* (c + <f)
+ 3ft (r + 2crf) } ,
"C = a" + 5fl^ (/' + r + ^/ + f) + 1(V { {b -]- r + df + 2 {h + c + d) e}
+ lOa^ ■; (ft + c)' + 3 (ft + c)- ((/ + ^) + 3 (ft + c) (<i* + 2<fe) }
+ 5a [ft' + 4ft=* (c + rZ + ^) + Oft- {{c + dy + 2 (c + d) e}
+ 4ft {c' + 3r (fZ + c) + 3c {(P + 2rf«) }] .
THfiORIE DES FUNCTIONS NUM£RIQUES SIMPLEMENT
PfiRIODIQUES. ,
Par Edouard Lucas, Professeur au Lycee Charlemagne, Paris.
(Voir pag. 240 et suIt.)
Section XXIV.
De V apparition des nombres premiers dans les series recurrentes de premiere espece.
Dans les series recurrentes de premiere espece, a et b designent deux
nombres entiers, positifs et premiers entre eux ; il est d'abord evident que
les diviseurs preniiers de a et de J, ou de Q zz «&, ne se trou vent jamais comme
facteurs dans la serie ; il ne sera pas tenu compte de ces diviseurs dans tout
ce qui va suivre. On deduit immediatement de la premiere des formules (4),
la demonstration du theoreme de Fermat. En effet, on a, en negligeant les
multiples de py suppose premier et impair,
2P-1 a^ — hj ^ gp-i (j^^^^ ^
a — b
Multiplions les deux termes de la congruence par 8=: a — 6, nous obtenons
2p-'{aP—b^) = {a — bYy (Mod.;?);
supposons a — 6 zz 2, et divisons par 2^""\ il vient
aP — b^ ^ a — ft , (Mod. p) ,
ou, encore
a^ — a=.b^ — 6 , (Mod. p) .
Ainsi, le reste de la division de a^ — a, par p premier, ne change pas lorsque
Ton diminue a de deux unites, et par suite de 2, 4, 6, 8, . . . unites ; mais pour
a = ou fl = 1, ce reste est nul ; done a^ — a est toujours diivisible par le
nombre premier />, quelque soit Tentier a. Par suite, si le nombre entier a
n'est pas divisible par p, la diflfierence a^""^ — 1 est divisible par p; c'est pre-
cisement I'enonce du theoreme en question.
En supposant maintenant a et ft quelconques, mais non divisibles par p^
les differences
a^-^— 1 et ftp-^ — 1
73 289
290 Lucas, ThSorie des Fonctions NumSriques Simplenient Periodiques.
sont divisibles par^; done, si a — b n'est pas divisible par^, on a
U,^,= '^ ^-— =0, (Mod.i>).
a —
Par consequent, les diflFerents termes des series recurrentes de premiere espece
contiennent, en exceptant les diviseurs de Q =: ab et de S zz a — 6, tous les
nombres premiers en facteurs.
Mais, s'il est vrai que p divise Up_i, on peut, dans la plupart des cas,
trouver un terme de rang inlerieur a /> — 1, et divisible par p, Designons
par 6) le rang d^arrivee ou d'apparition du nombre premier p dans la serie des*
Un\ il resulte des principes exposes (Section XI), que Ton a, pour k entier
et positif,
U^ = 0, (Mod.;>);
ainsi, tous les termes divisibles par p ont un rang egal a un multiple quel-
conque du rang d'apparition.
II resulte encore des principes exposes (Section XIII), que les termes,
dont le rang est un multiple quelconque de (p — Vjp^"^^ sont divisibles par
p^] mais il peut exister d'autres termes divisibles par p\ pour deux raisons
bien diiferentes ; 1° lorsque le rang d'arrivee w de p diflFere dep — 1; 2° lors-
que le nombre premier p arrive pour la premiere fois a une puissance supe-
rieure a la premiere ; mais, cela connu, il est facile de tenir compte de ces
singularites. En general, si m designe un nombre quelconque premier avec
Q, et <^ (m) Vindicateur de m, c'est-a-dire le nombre des entiers inferieurs et
premiers a m, on a la congruence
(135) £^^(m) = 0,* (Mod. m);
cette congruence correspond au theoreme de Fermat gSn^alise par Euler.
Inversement, si Ton a
Ur, = 0, (Mod. m) ,
on en deduit
n =: kfiy
fi designant un certain diviseur de <^ (m), et k un entier positif quelconque.
Les resultats que nous venons d'obtenir conduisent a la forme lineaire
des diviseurs premiers de U^. En effet, si o designe toujours le rang d'arrivee
dej?, on a, puisque Up_i est divisible par^,
P 1 = ^oO) ,
et, par suite
(136) p = k^ + l.
Lucas, Theorie des Fonctions Numeriques Simpletnent PSriodiques. 291
Nous appellerons diviseurs propres de ?7„ tous les facteurs premiers de U^, que
Ton ne rencontre pas dans les termes do rang inferieur, et diviseurs impropres^
les facteurs premiers contenus prealablement dans les termes de la serie. On
a alors les deux propositions suivantes :
Theorem E I : Les diviseurs imprapres des termes U^ des fonctions simplement
periodiques sont des diviseurs propres des termes dont le rang est tin diviseur de n.
Theorem E II : Les diviseurs propres des termes U^ des fonctions periodiques
de premiere espece apparUennent a la forme lineaire kn + 1.
Enfin, si Ton observe que Ton a trouve
on a encore :
Theoreme III : Les diviseurs propres de V^ appartiennent a la forme lineaire
2kn + 1.
On deduit encore de ce qui precede la demonstration du theoreme suivant,
qui n'est qu'un cas particulier du theoreme de Lejecjne-Dirighlet, sur les
progressions arithmetiques :
Theoreme IV : Quel que soit Ventier m, il y a une serie indefinie de nomhres
premiers de la forme lineaire km + 1.
En eifet, il est d'abord evident que, pour une valeur suffisamment
grande de n, le terme U^ possede necessairement un ou plusieurs diviseurs
propres de la forme kn + 1. Par consequent, si Ton fait successivement n
egal a
m, pm^ p^m^ p^m^ . . . p^m^
p etant premier, les termes correspondanls possedent tous, a partir d'un certain
rang, des diviseurs de la forme consideree ; le theoreme est done demontre.
II resulte encore, du theoreme I (Section XX), que ces diviseurs appar-
tiennent en outre aux diviseurs de la forme quadratique x^ ± py^, suivant
que Ton prend pour p un nombre premier de la forme 4^ + 3, ou de la forme
Les theoremes precedents permettent encore de determiner les diviseurs
des fonctions numeriques de premiere espece; nous donnerons d'abord les
deux exemples suivants dus a Euler.
ExEMPLE I : Soit, dans la serie de Fermat,
U^ = 2^—l = 18446 74407 37095 51615,
on a, d'apres les formules precedentes,
292 Lucas, Theorie des Fonctions Numeriques Simplenient Periodiques.
on a immediatement les decompositions en facteurs premiers
U, = l, Fi = 3, F2 = 5, V, = VJ, F8 = 257, Fi6 = 65637;
et
F32 = J2949 67297 .
Les diviseurs de V^ appartiennent a la forme lineaire Q^k + 1 ; en essayant
les diviseurs premiers de cette forme
193, 257, 449, 577, 641,
on trouve
F32 zz 641 X 67 00417. *
L'essai des diviseurs premiers de meme forme
641, 769, 1153, 1217, 1409, 1601, 2113,
et inferieurs a la racine carree du second facteur de F32, indique presque
immediatement que 67 00417 est un nombre premier.
Fermat avait annonce, mais sans dire qu'il en eut la demonstration,
dans une lettre du 18 Octobre 1640, que la formule 2'-"+ 1 donnait toujours
des nombres premiers. Cette formule se trouve en defaut, d'apres la decom-
position precedente, due a Euler, pour w zz 5.
On sait, d'autre part, que Gauss a demontre que Ton pent divisor la
circonference en 2^" + 1 parties egales, lorsque ce nombre est premier, et
seulement dans ce cas, par la regie et le compas. Nous indiquerons plus loin
une methode de recherche du mode de composition des nombres de cette
forme, basee sur la distribution des nombres premiers dans la serie de Pell.
Par la methode que nous venons d'exposer, en supposant que le nombre
2^" + 1 zz 18446 74407 37095 51617 ,
soit premier, il faudrait a un seul calculateur, pour le demontrer, tout en
profitant de la forme 128A: + 1, imposee aux diviseurs de ce nombre, environ
trois mille ans de travail assidu. f Par notre methode, il suflSt de trente heures^
pour decider si ce nombre est premier ou compose.
ExEMPLE II : Soit encore, dans la serie de Fermat, le terme
U,, = ^' — l = 21474 83647
dont le rang 31 est un nombre premier. Les diviseurs de Uzi sont, sans
exception, des diviseurs propres appartenant a la forme liHeaire 62^ -[- 1.
Mais, d'autre part (Section VJII, Theoreme I),^en tenant compte des formes
quadratiques de ses diviseurs, ou des formes lineaires correspondantes Sk' ^ 1,
* II est inutile, d'apros la loi de repetition, d'essayer 257 qui se trouve dans V^ . Nous avons demontre
quo les diviseurs de K32 apjjartiennent a la forme 128A + 1. (Aeademie de Turin, Janvier 1878 )
\ Aux mathematieiena de touies lea parliea du monde. — Oommunication stir la dicompoaUion des nombres en
leura Jacteurs aimplea. Par M. F. Landry. Paris, 1867. (Note da la page 8.)
Lucas, Theorie des Fonctions Numeriques Simpleinent Periodiques. 293
on voit que tout diviseur premier de U^i appartient necessairement a Tune des
formes lineaires
248^ + 1 , 248^ + 63 .
"Or, EuLER* nous apprend qu'apres avoir essaye tons les nombres pre-
miers contenus dans ces deux formes, jusqu'a 46339, racine du nombre 2^^ — 1,
il n'en a trouve aucun qui fut diviseur de ce nombre; d'oii il faut conclure
conformement a une assertion de Fermat, que le nombre 2^^ — 1 est un nom-
bre premier. C'est le plus grand de ceux qui aient ete verifies jusqu'a
present." (Legendre, Theorie des Nombres, 3® edition, t. I, pag. 229.
Paris, 1830.)
Exemple III : On connaissait, depuis quelques annees, un nombre pre-
mier plus grand que le precedent, indique par Plana, dans son Menwire §ur
la Theorie des Nombres^ du 20 Novembre 1859.f Soit, en eflfet
F^ = 3^ + 1 ;
ce nombre a tons ses diviseurs propres de la forme 58^* + 1 ; mais d'autre
part, ces diviseurs appartiennent a la forme quadratique x^ + 3y^, et, par
suite, aux formes lineaires 12k + 1 et 12k + 7. En combinant Tune de ces
formes avec la precedente, on trouve que les diviseurs de F29 sont de Tune
des deux formes
348A: + 1, ou 348A:+175.
Plana a ainsi trouve la decomposition
F29 = 2^ X 6091 X 28168 76431 ,
et verifie que le dernier facteur est premier. II a encore indique (loc. cit,
pag. 140 et 141) que le quotient
Q29 1
I ^ = 58 16133 67499 ,
2X 59 '
n'a pas de diviseur premier inferieur a 52259, et que le nombre 2'^ — 1 n'a
pas de diviseur inferieur a 50033. Ces trois assertions sont inexactes ; on a
329 _ 1 -, 2 X 59 X 28537 X 203 81027 ,
3^ + 1 = 2^ X 523 X 6091 X 53 85997 ,
2^ _ 1 - 6361 X 69431 X 203 94401 .
Nous ajouterons que Ton trouve encore dans la memoire de Plana, la
decomposition
2^^ — 1 =z 13367 X 1645 11353 .
* Letire d Bei^oulli^ on 1771, — Memoires de V Academic de Berlin, annee 1772, pag. 80.
•fMemorie della Reale Accademia delle Scienze di TorinOf 2* serie, t. XX, p. 139. Turin, 1863.
74
294 Lucas, ThSorie des Fonctions NumSriqties Simplement Periodiques.
ExEMPLE IV : Nous donnerons encore quelques exemples de decomposi-
tion de la fonction numerique
(2m)*» — 1 ,
qui joue un role assez important dans les congruences de degre superieur.
Nous avons trouve les resultats suivants :
1 = 13 X 81 08731 ,
1 = 3 X 6 X 70 27567 ,
1 = 3 X 7 X 11 X 19 X 61 X 261 X 1 52381 ,
1 = 41 X 401 X 2801 X 2 22361 ,
l = 3x7x67X3o3X 11764 69637 ,
1 = 23 X 89 X 28 54510 51007 ,
1 = 5^ X 7 X 13 X 23 X 73 X 79 X 349 X 577 X 601 ,
1 = 97 X 3 31777 X 11347 93633 ,
1 = 3' X 29 X 113 X 13007 X 35771 X 44 22461 ,
1 = 7^ X 19 X 29 X 12211 X 8 37931 X 519 41161 ,
1 = 11 X 13 X 31 X 67 X 271 X 4831 X 71261 X 6 17831 ,
dont nous donnerons plus tard I'application a de nouvelles recherches sur le
dernier theoreme de Fermat.
'14'
1
14'
+ 1
20'«
— 1
20'"
+ 1
22"
1
22"
+ 1
24'^
1
2412
+ 1
28"
1
30'*
1
^30'^
+ 1
Section XXV.
De V apparition des nombres premiers dans les series recurrentes de seconde et de
troisieme espece.
En designant toujours par p un nombre premier quelconque, on sait que
le reste de la division de A ^ par p est toujours egal a 0, a + 1, ou a — 1,
suivant que A est un multiple, un residu quadratique, ou un non-residu
quadratique de p. Nous considererons les cinq cas suivants.
Premier cas. p est un diviseur de P.
On a (Zj = P^ ^t par consequent tous les termes Z7„ de rang pair de la
serie sont divisibles par p; en designant par p^ la plus haute puissance de
p qui divise P, les rangs des termes divisibles par p^^*" seront tous les
multiples de 2/>'*.
Lucas, TMorie des Foiictions NuniM^ques Simplement Periodigues. 295
Deuxieme CAS. p est un diviseur de Q.
Nous avons, par definition,
2" V "A ?/; = (P + V A)» — (P — V A)» ,
2»F„ = (P + V A)" + (P — V A)» ;
t
on a done, en supprimant les multiples de Q, par le remplacement de A par
P^, les congruences
2~P U,= {P+ Py , (Mod. Q) ,
2rV^={P+PY, (Mod. Q),
ou, plus simplement,
(137) u^=pn-i^ V„=P\ (Mod. Q) .
Par consequent, Un et Vn ne sont jamais divisibles par Q ou par Tun de ses
diviseurs, puisque P et Q ont ete supposes premiers entre eux. D'ailleurs ce
resultat s'applique aux series de premiere et de troisieme espece ; lorsque Ton
a Q = i 1, comme dans les series de Pell et de Fibonacci, nous n'aurons
pas a tenir compte du theoreme precedent.
Troisieme cas. p est un diviseur de A .
Lorsque^ est un nombre premier diviseur de A, les formules (4) donnent
immediatement,
(138) . Up = 0, Vp = P, (Mod. p) .
et, par suite cette proposition :
Theoreme : Bans la sSrie TJ de seconde espece^ tout diviseur premier p du
determinant A est un diviseur de Up.
II resulte d'ailleurs des principes exposes precedemment, qu'un diviseur
premier impair p de A arrive pour la premiere fois, dans Upet a la premiere
puissance.
QuATRiEME CAS. A est residu qiuidratique de p.
En changeant, dans la premiere des formules (4), ^ en |> — 1, on a
et, en appliquant les resultats obtenus (Section XXI) pour les congruences du
triangle arithmetique, on a
2^'^Up^, = — \P^-^ + P^-^A + pp-W + . . . + PA^'i^'] , (Mod. p) ,
296 Lucas, Theorie des Foncfions Numeriqu^s Simpleinent Periodiques.
et, par suite
2P-^U,_, = -P^^-^^-, (Mod.p).
Mais on a, par le theoreme de Fermat, P''""^ ^ 1 , (Mod. |>), et, puisque A
est residu quadratique de p, il en resulte que ?7p_, est divisible par 2?- On a
done cette proposition, qui s'applique aux series de troisieme espece, en tenant
compte du signe de A :
Theoreme : I)ans la serie recurrente U de seconde ou de troisieme espece j
tout noinhre premier p^ qui admet A pour residu quadratique, divise le terme Up_i.
La seconde des forraules (4) donne
2^-'V,_r = P^-' + (i^ — y(| — ^ ) pP-3^ + . . . + A^"*^,
et, par suit^
2^-^Vp^, = P^-' + P^-^A + + A^"^, (Mod.^),
ou bien
2''-^F,_.^ -^^,_^ . (Mod.;,).
Mais on a, dans le cas present
pp + i = p2 ^j. A^^ = A, (Mod.i>);
done
2''-^Fp^, = l, (Mod.|>),
et finalement, en multipliant par 2 et appliquant le theoreme de Fermat :
(139) Fp_i = 2, (Mod.^).
CiNQQiEME CAS. A est nou-residu quadratique de p.
On a, comme precedemment,
JL • i^
et, puisque p est premier,
2U,^, = P{l + A-n,
p
2V,^, = p' + a:a^-
Mais, par hypothese A est non-residu quadratique dep, et, par suite
A^^" = — 1, (Mod.^);
Lucas, Th^orie des Fonctions Num^igy^s Simplement Periodigues. 297
on a done
(140) ir,^, = 0, V,^, = 2Q, (Mod.^));
de la, cette proposition :
Th£OR£ME : Dans les series recurrentes U^ de seconds et de troisieme espece^
tout nombre premier p^ dont A est un non-remlu qtiadratiquey divise C^ + 1 .
Designons encore par cj le rang d'arrivee du nombre premier^ dans la
serie des U,, , et par k un nombre entier quelconque ; on a
U^=0, (Mod.p);
par consequent, si jp n'est pas diviseur de Q ou de A, on a
A-oG) = i> ==F 1 ,
en prenant le signe — ou le signe + suivant que A est residu ou non-residu
de |> ; on en deduit
i> = *aw it 1 ,
et,'par consequent :
Th^oreme : Bans les series recurrentes de seconde espece^ les diviseurs propres
de U^ sont de la forme lineaire p =i ku ^ 1, suivant que A est rhidu ou non-
rSsidu de p.
En suivant une marche analogue a celle que nous avons suivie dans le
paragraphe precedent, on obtient par la consideration des diviseurs de U^ ,
le theoreme suivant.
Th6oreme : II y a une serie indefinie de. diviseurs premiers communs aux
formes quadratiques x^ — Qy^ et oc[ — pt^^ , hrsque p dSsigne un nombre premier
de la forme 4^ + 1 ; et une serie indefinie de diviseurs communs aux deux formes
^ — Qj{t ^^ ^^ + V}h > fe^^jw^ 'P designe un nombre premier de laform£ 4j + 3.
Nous appliquerons les resultats qui precedent, aux series de Fibonacci
et de Pell. Pour la premiere, on a P zz 1, Q = — 1, et A zz 5, d'autre part,
on sait, * que le nombre 5 est residu de tons les nombres premiers qui sont
residus de 5, et non-residus de tous les nombres premiers impairs qui sont
non-residus de 5 lui-meme. Par consequent:
•
Dans la serie de Fiboxacci, tout nombre p premier impair^ de la forme lOj i 1,
divise le terme de rang p — 1, et tout nombre p premier impair de la forme lOq i 3
divise le terme de rang p + 1.
D'ailleurs, les nombres 2 et 5 divisent respectivement les termes dont le
rang est un multiple de 3 ou de 5.
* QAVBa,—I>i8qui8itione3 ArithmeiUce. Nos. 121, 122 et 128.
75
298 Lucas, Theorie des Fonctions Numeriques Simplement Periodiques.
Pour la serie de Pell, Pzz2, Q = — 1, Aii:2^X2; d'autre part, on
sait que le norabre 2 est residu de tout nombre qui n'est pas divisible par 4, ni
par aucun nombre premier de la forme 8j + 3 ou 8j + 5, et non-residu de
tous les autres ; par consequent :
Dans la serie de Pell, tout nombre premier p de la forme 8j i 1 divise C^-i,
et tout nomhre premier p de la forme 8j i 3 divise U^ + i*
Les theoremes que nous venons de demontrer conduisent a la decompo-
sition des termes des series recurrentes de seconde et de troisieme espece, en
facteurs premiers. On a ainsi, par exemple, dans la serie de Fibonacci :
U,, = 1655 80141 = 2789 X 59369 ,
U^ = 5 33162 91173 = 953 X 559 45741 ,
U,, = 95 67220 26041 = 353 X 27102 60697 .
Nous ajouterons une remarque importante dont on retrouve I'origine dans
la correspondance de Fermat, mais seulement pour les series de premiere
espece.
Soit encore, par exemple, la serie de Fibonacci; les nombres premiers p,
des formes lineaires 20q + 13 et 20q + 17, divisent U^^i, et Ton a
;? + 1 = 20j + 14 ou;? + 1 = 20j + 18,
et aussi
mais, d'autre part, les diviseurs de F2n + i appartiennent aux formes lineaires
20q + 1, 9, 11, 19 ; par consequent, les nombres premiers de la forme 20q + 13
ou 20j + 17 divisent respectivement Uiog^7 etf/io^ + g, et disparaissent de la
serie des V^ qui ne contient done pas tous les nombres premiers. En appli-
quant ce raisonnement aux series de Fermat et de Pell, on en deduit les
principes suivants :
Dans la sSrie de Fibonacci, les termes F„ ne contiennent aticun nombre
premier des formes lineaires 20 j + 13, 20 j + 17.
Dans la serie de Fermat, les termes F„ ne contiennent aucun nombre premier
de la forme 8j + 7.
Dans la serie de Pell, les termes F„ ne contiennent aucun nombre premier de
la forme 8j + 5.
Nous donnons dans le tableau de la page 299, la decomposition en
facteurs premiers des termes de la serie de Fibonacci, limitee aux soixant^
premiers termes.
Lucas, Theorie des Fonctions NuinMques Simplenient Periodi^ues. 299
O CD 00 ^ Od 0<
l»k 00 lO •->
IsS
CCCQ0^O»Cni»k00h9»^
0<OQO^O)0«i»k00^9<-'
00 0« 00 •-• •-•
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300 Lucas, Theorie des Fonctiofis NumMques Sinvplemeni PModigues.
Section XXVI.
Sur la periodicite des fonctions numeriques et sur la generalisation
du Canon Arithmeticus.
Les resultats developpes dans les deux sections precedentes, conduisent
immediatement a la periodicite numerique des fonctions que nous etudions
ici, par la consideration de leurs residus suivant un module premier p ou
suivant un module quelconque 7n. Cette question a ete presentee sous une
forme differente, et seulement pour les series de premiere espece, par GausSi
dans les Disquisitiones AritJimetkas^ sous le nom de theorie des indices^ et deve-
loppee par Jacobi dans le Canon Arithmeticus. Tous ces resultats peuvent
etre resumes et generalises, dans le theoreme fondamental suivanti qui con-
tient une extension du Theoreme de Fermat genSralis^ par Euler.
ThIioreme fondamental : Si Von designe par m un nomber premier avec
le produit des racines d^une equation du second degri a coefficients comm^nsurables^
m zz p'r^sT . . . . ,
(A \ . ^~ *
— j le reste de la division de A ^ par p,
et egal a -\- 1 ou a — 1, suivant que A est rhidu quadratique^ ou non-rSsidu
quadradque de p ; et, soit, de plus
,W=,.-v-v-....[,-(|)][.-(f)][.,(4)]..,.
on a la congruence
(142) f7'^(^) = 0, (Mod. m).
Beciproquement, si U^ est divisible par m, le nombre n est un multiple quel-
conque d^un certain diviseur (i de ^ {m) .
Ce nombre fi est, par extension, Texposant auquel appartient a ou J par
rapport au module m ; on retrouve le theoreme d'EuLER, en supposant J = 1.
Quant a la periodicite numerique des residus, elle resulte des formulas
d'addition. On a d'abord, en faisant w =z ^u dans les formules (49) ,
2V^^j^=r^V^ + AU^U^;
par consequent, si q designe le rang d'arrivee du nombre premier p dans la
serie des ?7„ , on a
(143) ll-^'^^l^l-' \ (Mod.^,).
Lucas, Theorie des Fonctwns NumSriques Simplement Piriodigues. 301
Supposons d'abord qu'il s'agisse des fonotions de premiere espece, ou
lorsque A est r^sidu de j>, des fonctidns de deuxieme et de troisieme especo ;
determinons le nombre k de telle sorte que Ton ait
F^ = 2, (Mod.^),
ce qui a lieu pour kG>=:p — 1, mais aussi, dans la plupart deS cas, pour un
certain diviseur ndep — 1 ; on aura alors, pour h entier et positif, mais quel-
conque, les formules
(144) ^'"*"Z^- I (Mod.i,).
Gelles ci sont analogues aux formules qui donnent la periodicite des fonc-
tions circulaires; leur application conduit, lorsque Ton remplace le nombre
premier p par un module quelconque m; et que Ton tient compte de la, loi de
repetition^ a des formules nouvelles contenant la generalisation de resultats
indiques par Arndt et Sancery *
Mais dans le cas des series de seconde et de troisieme espece il n'en est
plus absolument de memts, lorsque A est non-residu de p. En posant
u' = ^ + 1, on a alors ;
V = ov \ ^^'^- P^ '
et, plus generalement, pour k entier et positif,
(i«) v:::=l'k:] <*'»^-'"
par consequent, si fi designe I'exposant auquel appartient Q suivant le module
Py on aura
Ainsi dans ce dernier cas, Tamplitude de la periode est 6gale a //u'.
Section XXVII.
Sur V inversion du thSoreme de Fermat et sur la vSrifi^ation des grands nombres
premiers.
On sait que le theoreme de Wilson qui consiste, pourp premier, dans la
congruence
1.2.3 (^ — 1) = — 1 , (Mod. p) ,
* Journal de Orelle^ t. xxxi ; pag. 260 et Buiv. 1846. — Bulletin de la Socieie Mathhnaiique de France^ t. lY.
pag. 17 et suiv. Paris, 1876.
76
302 Lucas, ThSorie des Fonctians Nunwriques Simplenient Periodigues.
s'applique exclusivement aux norabres premiers, et donne, par suite, un
precede theorique, mais illusoire dans la pratique, pour reconnaitre si un
nombre donne est premier. II n'en est pas de meme du theoreme de Febmat.
En designant par a un nombre inferieur a^, on a
a^-'^ = l, (Mod. p);
mais ce theoreme n'est pas restreint aux nombres premiers, et cette congru-
ence pent etre verifiee pour des modules composes ; ainsi, on a, par exemple,
237x73-1 = 1^ (Mod. 37X73).
Cependant, on pent enoncer le theoreme suivant que Ton doit considerer
comme la proposition reciproque de celle de Fermat.
Theoreme : Si (f — 1 est divisible par p, lorsque x =z p — 1, et rC est pas
divisible par p^ pour x inferieur dp — 1, le nombre p est premier.
On sait que, dans ce cas, a est une racine primitive de p; de plus, il est
facile de voir que si ^ — 1 est egal a une puissance de puissance de 2, a est
non-residu quadratique de ^. Ce theoreme centre dans le suivant, dont la
demonstration resulte immediatement des proprieties des fonctions numeriques
simplement periodiques, et s'applique aux trois especes de series :
Theoreme fondamental : Si dans Vune des series recurrentes U^ , le temie
Up^i est divisible par p, sans qu'aurcun des termes de la sM£ dont le rang est un
diviseur de p — 1 le soit^ le nombre p est premier ; de meme si Up + i est divisible
par p, sans qu'aucun des termes de la serie dont le rang est un diviseur de p-\-\
le soit, le nombre p est premier.
En effet, puisque p divise C^ ± i , tons les termes divisibles par p ont un
rang 6gal a un multiple quelconque d'un certain diviseur de ^ ± 1 ; d'autre
part, supposons p non premier et egal, par exemple, au produit de deux
nombres premiers r et 5 , on a
Ur±t = 0, (Mod. r), 17, + ! = 0, (Mod. «) ,
et, par suite le terme dont le rang est (r i 1) (^ ± 1) est divisible par rs ;
mais, par hypothese p divise le terme de rang rs i 1? ©t, par conesquent
aussi, le terme dont le rang est egal a la difference des precedents, c'est-a-dire
(r±l)(5±l)-(r5±l),
ou bien
±r±s±l±l.
Mais ce dernier nombre est evidemment plus petit que rs ; par consequent, si
p n'est pas premier, il divise un terme dont le rang est inferieur a ^ i 1 ;
c'est ce que ne suppose pas I'enonce.
Lucas, ThSorie des Fonctions NumSriques Simplement PSriodigties. 303
On obtiendrait le meme resultat en supposant p egal a un nombro
impair quelconque, en faisant voir (Section XXVI, Theor. fond.) que
mil — 4/ (m)
est plus petit que mil.
X>ans I'application de ce theoreme, on calcule les termes dont le rang est
un diviseur quelconque de ^ dh 1, au moyen des formules d'addition et de
multiplication des fonctions nuraeriques, que nous avons exposees ci-dessus.
Nous donnerons d'abord un exemple numerique tres-simple.
ExEMPLE: Soit
2^ — 1 = 127 .
Pour savoir si 127 est premier, nous calculous Ui2s dans la serie de Fibonacci ;
on a alors les formules
on forme ainsi le tableau
U, = U, {Vi-2)= U, X7,
U,, = U, (F/ — 2)= U, X47,
U,, = U,,{Vi — 2) = U,, X 2207,
Z7« = i7^(FJ — 2)= j;,, X 48 70847,
Um = U»{r^ — 2)= U»X 2732 51504 97407.
Or 127 divise le dernier facteur et ne divise aucun des precedents, ainsi
2732 51504 97407 = 127 X 18 68122 08641, par consequent 127 est un nombre
premier. On simplifie considerablement le calcul par la methode des con-
gruences, en rempla§ant continuellement les noinbres Fg, F4, Fg, . . . . par
leurs residus suivant le module 127. En tenant compte de cette observation,
le tableau precedent devient :
F4 = 3* — 2 = 7,
Fg = 7* — 2 = 47,
F,e = 47* — 2 = 48,
F3,= 48»-2 = 16,
Fe4 = 16» — 2 = 0.
Cette methode de verification des grands nombres premiers, qui repose sur le
principe que nous venons de demontrer, est la seule methode directe et pratique,
connue actuellement, pour resoudre le probleme en question ; elle est opposee,
pour ainsi dire, a la methode de verification d'EuLER, deduite de la consi-
(Mod. 127).
>^^^x
304 Lucas, Theorie des Fanctions Numeriques Simplement PSriodigues.
deration des residus potentiels, Dans celle ci, on divise le noinbre soup§onne
premier, par des nombres inferieurs a sa racine carree, et qui appariiennent a
des formes lineaires determinees que Ton doit d'abord calculer ; le dimdende
est constants et le diviseur variable^ mais inferieur, il est vrai, au nombre essaye ;
c'est Vinsucces de ces divisions dont le nombre est considerable, malgre la
forme lineaire du diviseur, qui conduit a aflfirmer que le nombre essaye est
premier. Dans notre methode, au contraire, on divise, par le nombre soup-
§onne premier, des nombres d'un calcul facile, obtenus par la multiplication
des fonctions numeriques; ici le dimdende est variable et le diviseur constant;
par consequent, on remplace les divisions par de simples soustractions, si Ton
a calcule prealablement les dix premiers multiples de ce diviseur constant ;
en outre, le nombre des operations est peu considerable; c'est le succes de
Toperation qui conduit a aflSrmer que le nombre essaye est premier. Ainsi,*
en cas de reussite, notre methode est aflfranchie de Tincertitude des calculs
numeriques.
Pour verifier la derniere assertion du P. Mersenne, sur le nombre sup-
pose premier
et qui a soixante-dix-huit chiffres, il faudrait a Thumanite tout entiere, formee
de mille millions d'individus, calculant simultanement et sans interruption,
un temps superieur a un nombre de siecles represents par un nombre de vingt
chiflfres ; par notre methode, il suffit d'eflfectuer successivement les carres de
250 nombres ayant 78 chiflfres, au plus ; cette operation ne demanderait pas,
a deux calculateurs habiles controlant leurs operations, plus de huit mois de
travail. Nous appliquerons d'abord le theoreme fondamental a la verifica-
tion des grands nombres premiers de la serie de Fermat qui appartiennent a
la forme
p = 2^ + «— 1,
dans laquelle nous supposerons I'exposant 4 j + 3 egal a un nombre premier
tel que 8j + 7 soit un nombre compose. En eflfet, si 4^ + 3 n'est pas
premier, le nombre p est compose ; d'autre part, nous avons demontre (Sec-
tion XXIII) que si 4 j + 3 et 8j + 7 sont premiers, le nombre p est encore
compose.
En supposant p premier, on a immediatement
^ = 2^ — 1 , (Mod. 6) ;
done, dans cette hypothese p est non-residu de 5, et divise le terme dont
rang est egal k p + 1 ou a I'un des diviseurs de |> + 1, dans la serie de
.^
Lucas, ThSorie dea Fonctions NumSriques SimpUment PSriodiques. 305
Fibonacci ; mais tous ces diviseurs sont de la forme 2^ , et pour former les
termes qui correspondent a ces rangs, il suflSt d'appliquer les formules de
duplication des fonctions numeriques. On a alors
et Tapplication du theoreme fondaraental donne le principe suivant:
Th:6oreme II : Soit le nombre p = 2** + * — 1 pour lequel 4^ + 3 est pre-
mier ^ ^< Sj + 7 compose ; on forme la serie r„
1, 3, 7, 47, 2207,
par la relation^ pour w > 1,
/•n + i = ^ — 2;
le nombre p est premier lorsque le rang du premier terme^ divisible par p, occupe un
rang compris entre 2j + 1 ^# 4j + 2; le nombre p est compost, si aucun des 4tq-\-2
premiers termes de la serie n'est divisible par p ; enfin^ si a designe le rang du
premier terme divisible par p^ les diviseurs de p appartienn^t a la forms liniaire
2*ir ± 1, combinee avec celles des diviseurs de x^ — 2^ .
Dans la pratique, on calcule par congruences, en ne conservant que les
residus suivant le module |>, ainsi que nous I'avons montre precedemraent
pour le nombre p z= 2^ — 1 . Nous avons indique un autre precede .de
calcul, qui repose sur I'emploi du systeme de numeration binaire, et qui
conduit a la construction d'un mecanisme propre a la verification des grands
nombres premiers.
Dans ce systeme de numeration, la multiplication consiste simplement
dans le deplacement longitudinal du multiplicande ; d'autre part, il est clair
que le reste de la division de 2r par 2** — 1 est egal a 2^, r designant le reste de
la division de m par n; par consequent dans Tessai de 2^* — 1, par exemple, il
suffira d'operer sur des nombres ayant, au plus, 31 chiflFros. Le tableau de la
page 306 donne le calcul duTesidu de Fae deduit du residu de F25 suivant le
module 2^^ — 1, par la formule
F^=(F^r-2, (Mod.2«^-l);
les Carres noirs representent les unites des diflferents ordres du systeme binaire,
et les Carres blancs representent les zeros. La* premiere ligne est le residu de
F25; les 31 premieres lignes numerotees — 30 figurent le carre de F25; les
4 lignes numerotees 0, 1, 2, 3 du bas de la page indiquent I'addition des
unites de chaque colonne, avec les reports ; on a retranche une unite de la
premiere colonne a gauche ; enfin la derniere ligne est le residu de F26 .
77
306 Lucas, Theorie des Fonctions Numerigues Simplemeni PSriediques.
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1
CaLCUL DU RESIDU DE F26 AU MOYEN DE F25 SUIVANT LE MODULE 2*^ — 1.
Le tableau de la page 307 contient Tensemble de tous les residus
de F2, F2) F3, . . . . Fm, F30 suivant le module 2'* — 1. La derniere ligne,
Z 2 8 2
entiereinent composee de zeros, nous montre que 2^* — 1 est premier.
Lucas, ThSarie des Fonctums NumSrigueB Sirrvplement PModiques. 307
80 29 28 27 26 25 24 28 22 21 20 19 18 17 16 15 14 18 12 11 10 9
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1
DiAORAMME DU NOMBRE PREMIER 2*^ — 1 .
Ce tableau est, en quelque sorte, un fragment du Cam)n Arithmeticas^
correspondant au nombre premier 2*^ — 1 pour la racine primitive — =-
On pourrait ainsi construire les diagrammed des nombres premiers 4e la
forme 2** + ' — 1. Nous donnons aussi celui du nombre 2^® — 1 ; nous espe-
rons donner ulterieurement ceux des nombres 2*' — 1 et 2^^ — 1.
306 Lucas, Theorie des Fonciiona NumSriguea Simplemmt PModigws.
18 IT 16 IS U 18 12
II 10
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DIAQBA.HUB DU NOHBBE FBEM'IBB 2'* — 1 .
Lorsque Ton aura, par rapplication du theoreme fondamental/A verifier
g» I I
de .grands nombrea premiers de la forme -^ , on emploiera aussi avec
o ± 1
succes le systeme ternaire, dans lequel on se aervira seulement des chiflfres
0, 1 et 1, a caracteristiques positives on negatives. Pour la verification des
grands nombres de la forme 10*" + 1 , on se servira facilement du systeme
d^imal, pour le calcul des r^sidus.
Lorsque le nombre eaaaye n>'est pas premier, nous avona vu qu'on ne
trouvera aucun residu nul. Soit, par exemple, le nombre p = 2" — 1 = 2047 ;
les r^sidus que nous considerons sont, dans ce cas,
1, 3, 7, 47, 160, 1034, 620, —438,-676, 160, ,
et se reproduisent periodiquement a partir de 160 suivant les cinq residus
160, 1034, 620, — 438, — 576.
On peut donner une autre forme d'enonee, au Theoreme II, et aux
suivants, en tenant compte des formules qui concernent les radicaux continus
(Section XV) ; on a, par exemple :
Lucas, Theorie des Fonctions NumSriqties Simpleinent Periodiques. 309
Th^osemb III : Pour que le nombrep z=2^^^ — 1 soit premier j ilfaut et il
suffit que la congruence 3^2 cos ^— ^ , (Mod. p)^
soit veriJUe^ apres la disparition successive des radicaux confenus dans la valeur du
cosinus. Nous demontrerons ulterieurement que cette condition est necessaire
et suffisante.
On observera encore que les nombres de la serie
1, 3, 7, 47, 2207,
appartiennent tous, a partir du troisieme, a la forme lineaire 5y + 2 ; mais,
d'autre part (Section VIII, Theor. II), les diviseurs de ces nombres appar-
tiennent aux formes lineaires
202 + 1,3,7,9;
par consequent, chacun des termes de la serie precedente contient un diviseur
premier de la forme 6j + 2 ; il en resulte immediatement cette proposition :
Theorem E IV: II y a une infinite de nombres premiers appartenant a la
forme linSaire 5q + 2.
On voit encore que les nombres de la serie ont la forme 8A + 7 ; mais,
d'autre part, la forme des diviseurs quadratiques indique que les diviseurs de
ces nombres sont de Tune des formes 8A + 1, ou 8A + 7 ; par consequent,
chacun des nombres de la serie contient au moins uu diviseur de la forme
8A + 7, et, par suite :
Th^greme V: II y a une infinite de nombres premiers appartenant a la forme
Unfair e 8A + 7.
Les theoremes suivants permettent d'arriver a un grand nombre de
theoremes analogues, qui sont des cas particuliers du theoreme fondamental de
Lejeune-Dirichlet, sur la progression aritjinietique. Nous devons observer
cependant, que les nombres de la forme 5j + 2 ne sont pas tous compris dans
la serie que nous considerons ici, et qu'il en est de meme dans tous les autres
cas. Ainsi les theoremes precedents different, au fond, des cas analogues de
la progression arithmetique. La methode que nous employons s'applique
d'ailleurs, tres-facilement, a la demonstration du theoreme general suivant :
Theoreme VI : Si A et Q designent deux nombres quelconques ^premiers entre
eux^ la serie
^OJ ^l> ^2> ^3> • • • • ^n >
78
310 Lucas, Theorie dea Fomtiona Numerigues Simplement PSriodiques.
dans laqueUe on a
contient comme divisevrs, des nombres premiers tous differents.
Les formules de multiplication des fonctious numeriques conduisent a des
resultats analogues.
Par des considerations semblables aux precedentes, on demontrera les
theoremes suivants.
Theobeme VII : Soit le nombre p = A.2' — 1 , et
1°,
2°,
3°,
4°,
q = 0,
- (Mod. 4), et
on forme les q premiers termes de la s4rie
J=3,
^ = 7,
A~\,
A = l,
ou
= 7,
= 3,
(Mod. 10) ;
par la relation de rdcnrrence
*'l > **2 > **8 > ♦'i J • • •
» +
r = ri-2,
en prenantpour ri et r^ les termes U^ et V^, dela s^rie de Fibonacci. Le nombre
p est premier, lorsque le rang du premier terme divisible par p est 4gal a q. Si a
designe le rang du premier terme divisible par p, les diviseurs de p sont de la forme
2'.A.k:^ 1 , combinee avec celle des diviseurs de a^ — 2if et de a^ — 2Ai^.
TiitoBiMB YIII : On obtient un thhreme semblable en prenant
p = A.2^+l,
avec les valeurs
1°, 2 = 0,
2°, q = l,
3°, q = 2,
4°, ? = 3,J
A=5, =3,1
y (Mod. 4), et "J J^' ou ^^'
A =5, =7,
(Mod. 10) ;
A = 5, =l,j
soit, par exemple, j) = 3 . 2" — 1 = 6143. On forme la serie des residus
4, 18, 322, — 749, 1986, 388, 3110, 3016, 4614, 499, ;
done p = 6143 est premier.
TnioREME IX : Soit le nombre
^ = ^,3» — 1,
avec les valeurs
1°, q = 0,] A = 4,
A = 6,
A = 2,
A^2,
2°,
0,1
1,
3°, q = 2,
4°, 2 = 3, J
(Mod. 4), et
ou
= 8.1
= 8,
= 6,
= 4,
(Mod. 10) ;
Lucas, ThSorie dea Fonctians NumSrigues Simpleinent PModiguea. 311
an forme les q premiers termes de la serie
^ly ^2} ^3> • •• I
par la formule de recurrence
r.^, = f5 + 3ri — 3,
deduite desformules de triplication^ avec les conditions initiales
dans la sirie de Fibonacxji ; le nombre p est premier lorsque le rang du premier
terme divisible par p est Sgal a q; si a disigne le rang du, premier terme divisible
par p, les divisetirs de p sont de la forme di^. A .k ■:^ 1 , combinSe avec celle des
diviseurs qtiadratiques correspondants.
ExEMPLE : Pour jp = 2 . 3' — 1 , les residua sont
2, 17, 1404, ;
done p = 4373 est un nombre premier, puis qu'il n'a pas de diviseur Inferieur a
sa racine carree.
Theorexe X : On a un theoreme analogue en supposant
p = A,3!'+ 1,
avec les valeurs
2 = 0,
? = 1,
2 = 2,
2 = 3, J
^=0, =8,1
MMod.4). et jZ^;ou;^;
(Nfod. 10) ;
^=0, =4, J
et la relation de recurrence
ExEMPLE : Pour ;> = 2 . 3^ + 1 , on a les residus
4, 19, — 67,669, — 212,0;
done p = 1469 est premier.
Theoreme XI : Soit le nombre
p = 2A.5^+li
on forme la serie limitee a q termes , r© , ri , r, , r, , . . . ,
par la relation de recurrence
r,^, = rl + 5ri + 5r,,
et lee conditions initiales
dim la serie de Fibonacci ; le nombre p est premier, lorsque le rang du premier
terme divisible par p est igal a q; il est composSj si aucun des q termM n'est divi-,
sible par p ; enfin^ si a dhigne le rang du premier residu nul, les diviseurs premiers
de p sont de Vune des formes 2A . 5*A: ± 1.
312 Lucas, Theorie des Fonctions Num^riques Siinplement Periodiques.
Section XXVIII.
Sar la division gSometriqae efo la circanference eii parties egales.
Dans la section precedente, nous n'avons considere que la verification
des norabres premiers par Teraploi da la serie de Fibonacci ; il est clair que
toutes les autres series donnent lieu a de semblables theoremes; par suite de
I'indetermination laissee a la sorame P et au produit Q des deux racines de
I'equation fondamentale, on pourra toujours s'assurer du mode de composition
d'un nombre j9, lorsque Ton connaitra Tune ou Tautre des decompositions de
p -\- 1 on de p — 1, en facteurs premiers. Nous donnerons encore I'applica-
tion du theoreme fondamental, aux nombres premiers dans lesquels on peut
diviser geometriquement la circonference, en parties egales.
La theorie de la division geometrique de la circonference, en parties
egales, a ete donnee par Gauss, dans la derniere section des Disquisitio7ies
Arithneticce. II est convenu que cette operation ne peut etre executee que
des trois manieres suivantes: 1° par I'emploi simultan6 de la regie et du
compas, comme dans la construction ordinaire du decagone regulier (Euclide);
2^ par I'emploi du compas sans la regie (Mascheroni); 3° par I'emploi de la
double regie, sans compas, c'est-a-dire d'une regie plate dont les deux bords
sont rectilignes et paralleles. Cette idee ingenieuse est due a M. de Coat-
PONT, colonel du genie.
Gauss a demontre que, pour diviser geometriquement la circonference en
N parties egales, il faut et il suffit que
-^ z= 2** . a, . fl^ . flrjb . . . . ,
(i etant arbitraire, Gi, aj, a^j . . . . des nombres premiers et diflferents, en nom-
bre quelconque, mais de la forme
On a, pour les premieres valeurs de w.
Go = 3, 01 = 5,02 = 17 ,0^ = 237 ,a^ = 65537 ;
mais (75 est divisible par 641 (Section XXVI), et ne peut etre compris dans
I'expression de 2f. II reste done deux questions importantes a resoudre :
1° comment peut-on s'assurer que a„ est premier? 2° existe-t-il une serie
indefinie de nombres premiers a^? Nous ne repondrons, pour I'instant, qu'a
la premiere question.
Lucas, Theorie des FonctUms Numeriques Simplement Periodiques. 313
Si flr„ est premier, le nombre Q est residu quadratique de a^; done, dans
la serie de Pell, F^^-i est divisible par flr„; raais a^ — la pour diviseurs les
nombres,
2,2^,2% 2";
on a done, par Tapplication du theoreme fondamental, et. par les formules de
duplication, le theoreme suivant :
ti
Theoreme I : Soit le nombre a„ = 2^ + 1 ; on forme la serie des 2^ — 1 termes,
6, 34, 1154, 13 31714, 17 7B462 17794, . . . ,
tels que ckacun d^eux est egal au carrS du precedent diminui de deux unites;
le nombre a„ est premier, lorsque le premier terme divisible par a„ est compris entre
Us termes de rang 2^ ~^ et^t" — 1 ; il est compost, si aucun des termes de la sSrie rCest
divisible par a^ ; enjin si a <C 2" ~ ^ designe le rang du premier terme divisible par
a,., les diviseurs premiers de a„ appartiennent a la forme lineaire
On obtiendrait un theoreme analogue pour Tessai des grands nombres
premiers de la forme, A .2^ + 1.
m
Le savant P. Pepin a presente a I'Academie des Sciences de Paris {Comptes
rendtiSy 6 AoM 1877), un autre theoreme pour reconnaitre les nombres pre-
miers a„, qui rentre dans notre methode generale. En effet, au lieu de nous
servir de la serie de Pell, nous pouvons employer beaucoup d'autres series
recurrentes, et ainsi la serie recurrente de premiere espece, dont les termes
sont donnes par Texpression
^'- a—b'
dans laquelle a et J designent deux nombres entiers arbitraires. En faisant
6 = 1 et a quelconque, on obtiendra un theoreme analogue au precedent ;
mais si, de plus, par la loi de reciprocite des residus quadratiques, on choisit
pour a un non-residu de a^ suppose premier, a = 6, par exemple, il est clair
que le rang du premier residu nul sera exactement egal a 2** — 1. De cette
fa§on, la forme ambigiie donnee a I'enonce de nos theoremes disparait, il est
vrai, et Ton obtient alors une condition necessaire et suffisante pour que a^ soit
premier. II serai t facile de tenir compte de cette observation, et de donner
une serie de theoremes analogues, dans la recherche de la condition necessaire
et suffisante pour qu'un nombre 2"a/) ± 1 soit premier, lorsque a designe un
produit de facteurs premiers donnes, etp un nombre premier arbitraire. On
a, par exemple, les theoremes suivants.
79
314 Lucas, ThSorie des Fonctions Numeriqv^es Simplenient Periodigues.
Theoreme II : Lorsque p = lOq -{- 7 ou p = lOq + 9 ^t ««^ nombre pre-
mier, le nomhre 2p — 1 eat premier si Fan a, dans la serie de Fibonacci,
t/p = 0, (Mod. 22> — 1),
et reciproquement.
Theoreme III : Lorsque 2> = 4j + 3 est vn nombre premier, le nombre
2p + 1 est premier si Von a, dans la sSrie de Fermat,
U, = 0, (Mod. 2p + l),
et reciproquement
Theorems IV : Lorsque j> = 4j + 3 ^3< un nombre premier, le notnbre
2p — 1 est premier si Von a, dans le serie de Pell,
C/; = 0, (Mod. 2p — 1),
et reciproquement.
On doit cependant observer que si la methode indiquee par le P. PfipiN,
conduit a une forme plus claire et plus precise de Tenonce, qui devient ainsi
semblable a celui du theoreme de Wilson, il est preferable de s'en.tenlr, dans
Tapplication, a la forme que nous avons adoptee. En eflfet, rapplication de
ces theoremes repose sur une hypothese, celle de considerer coinme premier
un nombre pris arbitrairement dans une certaine forme ; il est plus probable
de supposer, au contraire, le nombre comme compose, ainsi que semble Tindi-
quer I'assertion du P. Mersenne. Par consequent, au lieu de reculer la
verification, jusqu'a Textreme limite, par I'emploi des non-residus quadra-
tiques, il serait plus pratique, dans Texemple, de se servir de Tun des
^ (2"~^) nombres qui appartiennent a Texposant 2"~^, pour le module a^ sup-
pose premier; mais cette recherche directe est fort difficile. On s'assurera
cependant que, par le theoreme I, il suffit, pour demontrer que a^, Oa, a^,
sont premiers, d'executer respectivement 3, 6, 12, operations au lieu du nom-
bre 4, 8, 16, qui lui correspond dans Tautre methode.
Section XXIX.
Sur la verification de V assertion du P. Mersenne.
Nous avons indique la marche a suivre pour les nombres de la forme
2^^^ — 1 ; il nous reste a indiquer une marche analogue pour les nombres
de la forme ^ = 2*^ ^ ^ — 1, tels que
2«i— 1, 2^7_i^ ,2^7—1.
Lucas, Theorie des Fonctions Numeriqnes Simpleinent Feriodiques. 316
En supposant p premier, — 1 est non-resida de p puisque p est de la forme
4:k + 3, et 2 est residu de j?, puisque p eat de la forme 8A: + 7 ; done — 2
est non -residu de p^ Par consequent, la serie conjuguee de celle de Pell,
c'est-a-dire la serie provenant de Tequation,
ar^ = 2a: + 3,
dans laquelle
Pz=2, Q = -3, A = 2^X(-^),
est propre a la verification des nombres premiers que nous considerons, puis-
que, si p est premier, Up^i est divisible par p. Les divi^eurs de ^ + 1 repre-
sentent toutes les puissances de 2 jusqu'a Texposant 4j + 1 ; il suffira done
de caleuler les residus de
par les formules ordinaires de duplication.
Mais nous devons encore faire une observation importante, au point de
vue du calcul. Puisque Ton emploie la formule
il est bon, si Ton efifectue le calcul des residus dans le systeme de numeration
decimale, de supposer Q = j3l,ouQ = ±10'*; car sans cela, on double la
longueur des calculs, ainsi qu'il est facile de s'en apercevoir; si Ton opere
dans le systeme de numeration binaire, il sera commode de supposer Q egal,
en yaleur absolue, a I'unite ou a une puissance de 2.
II est done preferable d'employer la serie recurrente provenant de
I'equation
ar^== 4r — 1,
dans laquelle
a = 2 + V"3 , b = 2 — V 3",
et
P = 4, Q=l, A = 2«X3.
En supposant que j> = 2**^^ — 1 est un nombre premier, on a, par la loi de
reciprocite,
(f ) = - (f ) .
puisque jp et 3 sbnt tous deux des multiples de 4 plus 3; d'autre part, par le
theoreme de Fermat
2^ + ^ — 1 = 1, (Mod. 3);
done 3 est non-residu de^, suppose premier, et, dans ce cas, Up^i est divisible
par^. Les diviseurs Aep + 1 sont egaux a toutes les puissances de 2 jusqu'a
4j + 1, et, de plus, Q = 1.
316 Lucas, ThSorie des Fonctions NumMques Simplement PSriodiqnes.
Par consequent, on formera la suite des residus
4, 14, 194, 37634,
tels que chacun d'eux est egal au carre du precedent diminue de deux unites.
ExEMPLE : Soit le nombre 2^^ — 1 = 8191 ; on trouve les residus
4, 14, 194, 4870, 3953, 5970, 1857, 36, 1294, 3470, 128, ;
done le nombre 2^^ — 1 est premier.
On a done le theoreme suivant :
Tu^OREME : Soit le noinhre ^ = 2** + ^ — 1 ; on forme la serie des rhidus
4, 14, 194, 37634, ,
tels que chacun d'eux est egal au carri du precedent diminuS de deux unites \
le nombre p est compose, si aucun des 4j + 1 premiers rSsidus liesi egal a ; fo
nombre p est premier si le premier residu nul occupe un rang compris entre 2q et
4^ + 1 ; si le rang du premier residu est 6gal a a <C2q, les diviseurs de p appar-
tiennent a la forme lineaire
2r^'k + l.
On aurait encore des theoremes analogues pour les nombres de la forme
Avant de terminer ce paragraphe, nous ferons observer que nous pensons
n'avoir qu'effleure le sujet qui nous occupe. II reste a trouver, comme pour
les nombres premiers, un criterium des nombres composes, affranchi de Tincer-
titude des calculs numeriques ; dans un grand nombre de cas, lorsque le nom-
bre essaye n'est pas premier, il se presente une periode dans la suite des
residus ; mais, s'il est vrai, comme nous I'avons demontre (Section XXVI),
que cette periode existe, lorsque I'on considere I'ensemble des residus de tous
les termes de la serie recurrente, il n'est pas demontre que cette periode se
manifestera, si Ton ne considere qu'un certain nombre d'entre eux, dont les
rangs sont en progression geometrique. C'est la un probleme important a
resoudre.
En second lieu, lorsque Tensemble des calculs demontre que le nombre
essaye n'est pas premier, peut-on arriver facilement, par la connaissance de la
serie des residus calcules, a la decomposition du nombre que Ton avait sup-
pose premier? Ces residus forment, comme nous I'avons dit, un fragment
d'un Canon Arithmeticus generalise, que Ton pent comparer aux tables des loga-
rithmes des sinus et des cosinus, ainsi que Ton compare le Canon Arithmeti-
cus lui-meme, aux tables des logarithmes des nombre^. C'est la un second
probleme a resoudre.
Lucas, Theorie des Fonctions NumSriqaes Simplement Periodiques. 317
Nous avons encore indique (Sections IX et XX I), une premiere generali-
sation de ridee principale de ce meraoire, dans Tetude des series recurrentes
qui naissent des fonctions symetriques des racines des equations algebriques
du troisieme et du quatrieme degre, et, plus generalement, des racines des
Equations de degre quelconque, a coefficients commensurables. On trouve, en
particulier, dans I'etude de la fonction.
^^ _ A (a** , ft** , e?" , . . . . )
**"" A (fl, ft, c, . . . .) '
dans laquelle a, ft, c, ... • designent les racines de I'equation, et A (a, ft, (?,... .)
la fonction altern^e des racines, ou la racine carree du discriminant de Tequa-
tion, la generalisation des principales formules contenues dans la premiere
partie de ce travail.
Enfin, il reste a developper la theorie de la division des fonctions nume-
riques, et son application a I'analyse indet^rminee du second degre et des
degres superieurs ; c'est une etude que nous esperons publier prochainement.
Nous donnons d'ailleurs, dans le dernier paragraphe qui suit, une autre
generalisation des fonctions numeriques periodiques, deduite de la considera-
tion des series ordonnees suivant les puissances de la variable.
Section XXX.
Sur la pSriodicite numSrique des coefficients differentiels des fonctions rationnelles
d^exponentielles.
m
L'etude des nombres premiers contenus dans les denominateurs des coef-
ficients des puissances de la variable, dans les developpements en series, lors-
que Ton suppose ces coefficients rediiits a leur plus simple expression, a conduit
EiSENSTEiN a la decouverte d'un theoreme remarquable. En effet, ce theoreme
fournit un criterium qui permet de decider, a la seule inspection des facteurs
premiers du denominateur, si la fonction qui represente la somme de la serie
supposee convergente, est alg^brique ou transcendante.
On salt encore que I'etude des facteurs premiers contenus dans les nume-
z""
rateurs des coefficients J?- de - — -— dans le developpement de , ou,
1.2.3. ..n ^^ 1 — e"
en d'autres termes, dans les numerateurs des nombres de Bernoulli, a con-
duit Oauchy, mm. Genocchi et Kummer, a d'importants resultats sur la
theorie des residus quadratiques, et sur celle de Tequation indeterminee
x^ +yp + zP = 0,
-D2n -"2*1
318 Lucas, Theorie des Fonctions Numhiques Simplenient PSriodigues.
dont Fermat a aflSrrae Timpossibilite en nombres entiers, j^our p > 2. Ainsi
M. KuMMEB a demontre que cette equation ne peut etre verifi6e par des
nombres entiers, lorsque p ne se trouve pas comme facteur dans les numera-
teurs des nombres de Bernoulli -82? -^4 1 -^e, . . . jBp_8.*
En se plagant a un point de vue different, MM. Clausen et Staudt ont
donne pour ces nombres cette expression remarquable
1 1 1^_ _ 1
2" ~ "a (3 • • • X '
dans laquelle Jin est un nombre entier, et les denominateurs 2, a, /?, y, X,
tons les nombres premiers qui surpassent d'une unite tons les diviseurs de 2/i.
Cette formule conduit an procede le plus rapide pour le calcul de ces nombres ;
M. Adams vient de donner, par son emploi, les valours des 62 premiers nom-
bres {British Association, — Plymouth, AoAt 1877.)
Nous avons indique aussi comment Tapplication combin6e des theoremes
de Fermat et de Staudt conduit a cette propriete que les nombres
a(a2»— l)J?2n,
sont entiers, quel que soit Tentier a. Nous allons montrer que Tetude des
coefficients de - — rr-— dans le developpement des fonctions rationnelles
1 .2 . 3.. . n
d'exponentielles, ou, en d'autres termes, les coefficients' differentiels de ces
fonctions, pour x = 0, conduit a des proprietes importantes.
On sait, en effet, que si Ton remplace x par les nombres entiers consecu-
tifs dans la fonction
^ (x) =Aa' + Bb'^ C(f+ 1)0' +
dans laquelle A, B, C, D, . . . . et a, ft, (?, rf, ; . . . sont entiers, on a, pour^ pre-
mier et k entier quelconque, la congruence
(148) ^lx+ k {p — l)2 = ^ (x) , (Mod. p).
Nous av'ons etendu cette propriety aux fonctions num^riques U^ et F„ ; il est
facile de voir que cette proposition s'applique aux coefficients diflferentiels
d'une fonction entiere de e' et de ^~*. II nous reste a montrer que cette
*Noufi avons mt diQe les di verses notations qui concernent ces nombres. La prdsente notation se prdte
beaucoup plus facilement aux doveloppements que comporte la tb^rie de ces nombres. Voir, sur ce sujet, lea
Notes ins6r6es dans les Compiea rendns de VAcadtmie des Sciences de Paris (Septembre 1876), dans les Annali di
Matematiea {2^ B^Tie^ tome Yin), dans les Nouvelles Annalea de Mathemaiigues (2* s^rie, tome zvi, pag. 167),
dans la Nouvelle Correapondance Mathematique (tome ii, pag. 328, et tome ill, pag. 69), dans The Mes9enger'of
Mathernaties, (Octobre 1877), etc.
Lucas, Th4orie des Fonctions NumMques Simplement PSriodigues. 319
proposition s'applique encore aux coefficients differentiels d'une fonction
rationnelle de 6* et de ^"".
Soit d'abord
(149) s6c ^ = 1 + a^ai^ + ^41?* + Og^ + ....;
M. Sylvester a appele nombres Euleriens {Comptes rendus, t. LII, pag. 161),
les coefficients, pris en valeur absolue, determines par la relation
(160) E,.z=( — 1)^,2,3.,.. (2n) (L,
on a, par le changement de x en a; V — 1 , la formule symbolique
(151)
u =
^ + e
— = e^.
dans laquelle on remplacera, dans le developpenient du second membre les
exposants de ^ par des indices ; ainsi
dxo
31: -cL .
En chassant les denominateurs de Tidentite (151). on obtient, par I'identifica-
tion des coefficients de jf, la formule •
(152) {E + 1)»+ {E — iy=z 0,
qui permet de calculer les nombres Euleriens par voie recurrente. On a
aussi le determinant
1 1. ....
(153) E^={ — iy
16 10
1 15 15 1
Co,
Ctt,
1 ^ (^2n
ce determinant est forme par les lignes de rang pair et les colonnes de rang
impair du triangle arithmetique. Les nombres Euleriens sent entiers et
impairs; Sherk a demontre qu'ils sent termines alternativement par les
cbiffres 1 et 5*. Ces proprietes sont des cas particuliers des suivantes.
En tenant compte des resultats obtenus (Section XXI), sur les congru-
ences du triangle arithmetique, la formule (152) donne, pour p premier, et
n=p — lj
(154) E^_^ + E^^, + E,^, + ... + E^ + E, = 0, (Mod.;?);
on a done cette proposition :
* Journal de OrelUj t. 79, pag. 67.
320 Lucas, Theorie des Fonctions Numeriques Simplement P^riodigues.
TnioRiMB : Si p est un nombre premier y la somme des nombres EuUriens^
pris avec les signes alternes -\- et — , dont Vindice est plits petit que p, eat divisible
par p.
Les premieres valeurs de E sont donnes par les relations
E, + Eo =
E,+ 6J52 + -Bo =
E, + 15E, + 15^2 + ^0 =
on a ensuite, par congruence, a partir de Ej^^u pour le module premier p
E,^, + Eo =
' '^p + 3 + 3-2J, ^ 1 + SE2 + Eq^O
E,^, + lOE,^^ + bE,^^ + bE, + 10^2 + ^0 =
la comparaison des deux groupes de formules qui precedent, donne successive-
ment
i^ + i — JEj, (Mod.p),
(165) ij, + 8 ^ ^4 ,
U
• • •
On obtient de meme, en general, pour k entier quelconque
(156) ^^^,(^.i) = i:2n, (Mod.i>),
et, par suite :
Th^gbeme: Les rhidvs des nombres EulSrienSy suivant un module premier
quelconque, se reproduisent pModiquement dans le nieme ordre, comm^ les rSsidtis
des puissances des nombres entiers.
Posons maintenant
2 \* ^ , ^ X . ^ x"
"•=Cm^-) =^^' + ^^'y +
E.
0.2
1.2
+ ■ . . + -B^
of
"1.2. ..n
+ ...
ou, sous la forme symbolique
(157) vr = «*.' ,
les coefficients E^ „ sont determines par la relation symbolique
(158) E^^ = {E'+E-+ E"+ + i:«]»,
dans le developpement de laquelle on remplace les exposants de E', E", . . . E^'^
par des indices, et en supprimant les accents. On a aussi la formule
Lucas, ThSorie des FoncUom NumSrigues Simplement Phiodigues. 321
(159) J5:.,, = [i:_i+^i]%
dans laquelle en remplace les exposants de -E._i et de J?, par des seconds
indices. Ces nombres E^^ que nous appellerons les nombres Euleriens d^ordre a
sont en tiers pour a entier et positif ; on demontre, comrae ci-dessus, que leurs
residus suivant un module premier se reproduisent periodiquement, et que
Ton a encore
(160) E^^=E^,^,,,_,^, {Mod.p).
Ces considerations s'appliquent, en general, aux coefficients diflFerentiels
d'une fraction ration nolle de ^, mais, dans certaines conditions, comme dans
le cas de
4) (^) •
Cependant, lorsque ^ (1) est nul, comme dans le developpement de qui
1 — ef
contient les nombres de Bernoulli, ce theoreme ne se presente plus imme-
diatement, puisque les coefficients ne sont plus entiers, et contiennent en
denominateur une serie indefinie de nombres premiers. Alors, on les mul-
tiplie par d'autres fonctions telles que
a {or— I)
afin de les rendre entiers, et d'appliquer les resultats qui proviennent des con-
gruences du triangle arithmetique.
Paris, D^cembre, 1877.
81
ON THE TWO GENERAL RECIPROCAL METHODS IN
GRAPHICAL STATICS.
By Henry T. Eddy, Cincinnati, Ohio.
§1.
The methods employed in the geometrical or graphical solution of stati-
cal problems are, in fact, merely applications of the parallelogram of forces,
so systematized and combined that the skilful draughtsman is able, by these
geometrical processes alone, to make computations suflSciently exact for prac-
tical purposes with a rapidity and insight into the real relations of the quan-
tities treated which often far surpasses that of any algebraic or numerical
process.
These geometrical processes are of two principal kinds. The first kind
determines the stress in each piece of any given framed structure, in which
the stresses are determinate, by drawing a reticulated polygon whose lines
represent these stresses and the applied forces. If a certain order of pro-
cedure be observed in drawing this reticulated force polygon, the frame and
force polygon stand in a reciprocal relationship to each other, which has
been clearly set forth elsewhere.* The second kind of geometrical process
enriployed aims at somewhat more general relations than those obtained by
the force polygon, and applies not only to any framed structure considered
as a single elastic piece of material, but to any elastic piece, framed or not.
The greater generality attained by processes of the latter kind is due to tho
assumption of such arbitrary forms of framing that their geometrical proper-
ties are of material assistance in determining the magnitudes sought.
Hitherto, one process only has been known which possesses this charac-
teristic generality, which process is based upon the properties of the equili-
brium polygon. An equilibrium polygon is often called a catenary or funicular
polygon, since it is the form assumed by a perfectly flexible string under the
action of the forces applied to it. Some of its geometrical properties have
»
■ - ~ ~ — ■!■ y --■ ■ -l...l». ■»■■■■■ ■^■■■» I ■■»—■— — _
♦Reciprocal Figures. James Clerk Maxwell. Philosophical Magazine, vol. 27. London, 1864.
Le figure reciproche nelle statica graflca. Lnigi Cremona. Milano, 1872.
A New General Method in Graphical Statics. Henry T. Eddy. Van Nostrand's Engineering Magazine,
vol. 18. New York, 1878.
322
EQUILIBRIUM POLYGON METHOD.
Fig. 1.
Eddt, On ihe Tvm Qeiural Reciprocal Mcthcde m Graphical StaUcc
/
Xf
FRAME PENCIL METHOD.
Fig. 3.
Eddt, On the Too Oeneral BeeiproceU Methods in Grapliical Statins.
Eddy, On the Two General Beciprocal Methods in Graphical Statics. 323
been long known, but the systematic development and practical application
of the geometrical relations involved in the equilibrium polygon and its
reciprocal force polygon is comparatively recent,* and has been principally
dependent upon the growth of modern higher geometry.
Culmann is justly regarded as the father of graphical statics as a prac-
tical method, and to him is due the establishment of the generality and
importance of the equilibrium polygon method. Mohr discovered an impor-
tant extension of the method f in showing that the deflection curve of a
straight elastic girder is a second equilibrium polygon, and the author has
further extended the method by showing its applicability to any curved elastic
girder or arch. J Many other writers have added to the subject and simplified
it. The bibliography of the subject will be found in suflScient detail else-
where. §
In a paper by Poncelet, || as given with some modifications by Wood-
bury, ^ the germs of a secQjid fundamental method appear, which is of the
same general nature as that of the equilibrium polygon to which it bears a
certain kind of reciprocal relationship ; but neither the author nor any sub-
sequent writer seems to have seen the possibility of establishing a general
graphical method of which this special solution would be a particular case.
The purpose of this paper is to establish the general properties of this
new general method, and to point out the reciprocity existing between it and
that of the equilibrium polygon. In attempting this, it seems best to estab-
lish the general properties of the equilibrium polygon from mechanical con-
siderations, instead of deriving them from higher geometry, and then to
obtain the corresponding properties of the new method, which we have
ventured to call the frame pencil method for reasons which will appear
subsequently.
§2.
In Fig. 1, let the forces which it is proposed to treat lie in one plane and
have their lines of action along the arbitrarily assumed lines, on each side of
* Grnphische Statik. C. Culmann. Zurich, 1866. Alio, vol. I, 2d ed. Zurich, 1875.
f Beitrag zur Theorie der Holz- und Eisenconstructionen. Zeitschr. d. Hannov. Ing. und Arch. Ver. 1868.
J New Constructions in Graphical Statics. Henry T. Eddy. Van Nostrand's Engineering Magazine.
Vol. 16. New York, 1877.
§ Ueber die Graphische Statik. J. I. Weyrauch. Leipzig, 1874. Translated as an Introduction to The
Elements of Graphical Statics. A. J. Du Bois. New York, 1876.
Lezioni di Statica Grafica. Antonio Favaro. Padova. Of this work there is a forthcoming French
translation. Gauthier-Villars. Paris.
II Memorial de I'officier du genie. No 12.
1[ Stability of the Arch. D. P. Woodbury. New York, 1858.
,^
v^
V
•-
324 Eddy, On the Two General Reciprocal Methods in Graphical Statics.
which are the letters ab, bc^ cd, de respectively : these lines may be called a
diagram of the forces, and this kind of notation is employed for convenience
of clearly showing the reciprocity between the diagram occupying the left of
the figure and the force polygon on the right, which we shall now construct.
Draw the lines at the extremities of which ai:e ab^ be, cdy de, in such direc-
tions and of such lengths that they shall severally represent, on some con-
venient scale, the magnitudes as well as the directions of the forces. This will
constitute the polygon of the applied forces. Since the force diagram shows on
some convenient scale of distances the relative position and direction of the
forces, and the force polygon shows their relative magnitude and also their
directions, it is evident that aft, etc., of the diagram is parallel to ab, etc., of the
polygon, and the number or position of the forces which can be treated is in
no way restricted to those we have arbitrarily chosen to illustrate the method
It is necessary, in drawing the force polygon abcde, to have regard to the signs
of the forces, so that, in passing continuously al^g the polygon, the motion
should be all in the direction in which the forces act, and not partly with and
partly against the forces. The arrow-heads show the sense in which each
force is taken, and the polygon is taken in the same sense by passing continu-
ously from e to a.
Next, assume any pole p as the common point of the rays pa,pb, etc., of
the force pencil p-abcde. The lengths of the rays pa, etc., represent, on the
assumed scale of forces, the magnitudes of the stresses in the members of a
irame of which we shall immediately draw a diagram ; and the directions of
pa, etc., show what directions the members of the frame must have. On
the left, draw the line lying between the letters pa parallel to the rB.y pa: —
this line, as will be seen later, is one side of an equilibrium polygon, and so it
will be called the side pa to distinguish it from the ray pa. The actual posi-
tion of the side is of no consequence ; its parallelism to the ray is the only-
important consideration. From the point at which the side pa intersects the
diagram of the force ab, draw the side pb parallel to the ray pb : and from the
intersection of the side pb with the diagram of be draw the side pc parallel to
the ray pc. In the same manner draw a side parallel to each of the rays, so
that finally the polygon p - abcde has a side parallel to each of the rays of
the pencil p-abcde; and the two are said to be reciprocal figures. The
reciprocity is that usually existing between a frame and its force polygon,
and it is that pointed out in connection with the kind of graphical process
first mentioned.
Eddt, On (he Two General Recip-ocal Metfiods in Graphical Statics. 325
. Consider the forces acting at the point pab of the diagram, supposing
that two members pa and pb of a frame meet here in a perfectly flexible hinge
joint and hold the force ab in equilibrium. From the parallelogram of forces
it is known that the sides of the triangle pab represent the relative magnitudes
of the forces in equilibrium at the joint jpoft. Similarly, the sides of the trian-.
glepbc represent the relative magnitudes of the forces held in equilibrium at
the hinge joint pic ; and in the sam% manner the forces meeting at the succes-
sive joints are represented in relative magnitude by the sides of the triangle
denoted by the same letters.
The sides pa, pb, etc., together, form an equilibrium polygon, so called
because such a frame requires no internal bracing in order to sustain the
applied forces. In the case we have taken, it is evident that the stresses are
all compressive, and the frame would form an equilibrated arch.
In the system of notation here used, p is regarded as denoting, on the
left, the area enclosed by the equilibrium polygon, b as the area between the
lines ai, pb and cb, and c as that between be, pc and dc, whether these converge
or diverge, i. e., whether the space so bounded is finite 6r not.
Close the force polygon by drawing the side ae, then ae represents the
relative magnitude of the resultant of the applied forces,. or the force which
will hold them in equilibrium according to the sense in which it is taken; for
forces proportional to the sides of a closed polygon have no resultant. The
side ae, also, is in the direction of the resultant, the only remaining question
being as to its position. Prolong the first side pa until it intersects the last
aide pe, and through this intersection draw the diagram of a force ae parallel
to the closing side ae of the force polygon. The diagram ae gives the true
position of the line of application of the resultant; for, suppose the prolon-
gations of pa and pe to be members of the frame with a hinge joint at 'their
intersection, then the sides of the triangle pas represent the relative magni-
tudes of the forces acting at this joint, when a force is applied at this point
which will hold the other applied forces in equilibrium.
Draw any line intersecting the sides pa and pe as 2 3, and also pq in the
force polygon, parallel to 2 3; then may the points 2 and 3 be taken as fixed
points of support of the equilibrium polygon or arch to which the forces are
applied. This arch has the span 2 3 and the applied forces cause a thrust
along 2 3, whose magnitude is given by the length of pq. This thrust along
pq may be sustained by a member joining 2 3 or by these joints in virtue of
^ .
326 Eddy, On the Two General Beciprocal Methods in GrapJncal Statics.
their being fixed. They, in either case, together sustain the resultant which
is divided so that 2 and 3 sustain aq and qe respectively, as clearly appears
fronn the fact that the triangles paq and qep represent the forces in equilibrium
at 2 and 3 respectively. The line 2 3 is called the closing line of the polygon p.
Again, choose any pole p^ as the common point of the force pencil
jp' - abode. To avoid multiplying lines p' has been taken upon pq prolonged.
Draw the equilibrium polygon p' whose stdes are parallel to the rays of the
pencil p' - abcde^ and to still further avoid the multiplication of lines, let the
side pa pass through the point 2. The sides of the polygon p^ are all under
tension except p'c^ but they might all have been either in tension or all in
compression had the side pa been made to pass through some point other
than 2.
As shown in connection with the polygon p the first and last sides of any
polygon must intersect on the diagram of the resultant, whence pa and pe
intersect on the line ae already drawn.
Prolong the corresponding sides pa and p'a, pb and pb^ etc., until they
intersect at 2, 1, etc., Ihen are the points 12 3, etc., upon one and the same
straight line. For, suppose the forces which are applied to the polygon
p to be reversed in direction, then the system applied to the polygons p
and p must together be in equilibrium ; and the only bracing needed is
the common member 2 3 parallel to ppy since the forces applied to p pro-
duce a thrust pq along it, and those applied to y a thrust qp\ while the
reactions aq and qe at 2 and 3 are in equilibrium. A similar result holds
for each of the forces separately ; e. g. the opposed forces ab acting on p and
p' may be considered to \>e applied at opposite points of a quadrilateral whose
remaining joints are 1 and 2 ; the force polygon corresponding to this quad*
rilateral is apbp ; hence 1 2 is parallel to pp. But 2 3 is parallel to pp^
therefore 12 3 are in one and the same straight line. The same may be
shown respecting the remaining intersections of corresponding sides. The
intersection of pc and pc does not &11 within the limits of the figure.
The properties which have been established with regard to the relations
of force diagram and equilibrium polygon to the force polygon and force
\\e\\c\\ are really of a geometric nature, and are not dependent upon the fact
that they represent relationships between forces. The proposition may be
stated in geometric language thus: from any points abed, etc., draw lines
converging to a single point as pole, also from the same number of points
Eddy, On the Two Grerieral Reciprocal Methods in Graphical Statics. 327
12 3 4, etc., lying in a straight line ; draw lines so that there shall be dne
line from each of the points 12 3 4, etc., parallel to each diflferent convergent
through abcdy etc.; then, if the common point of the convergent lines be
removed parallel to the line 1 2 3 4, so that the convergents severally revolve
about the points abcd^ etc., and the lines from 12 3 4 which are respectively
parallel to the convergents also revolve severally about 12 3 4, etc., the loci
of any and all of the intersections of these last mentioned lines are straight
lines, which are parallel to the lines joining the points aft, bc^ ac, etc. ; and
conversely, if these loci are the last mentioned straight lines, they revolve
about the fixed points 12 3 4, etc., which He in one and the same straight line.
For convenience of reference, we have here collected the special terms by
which the various parts of the reciprocal figures in the left and right are
designated.
abcdCy
p - abcde^
2 3 II pq,
ae.
Id Direction
and
Position.
^ Force Diagram,
Equilibrium Polygon,
Closing Line,
^ Diagram of Resultant,
Force Polygon.
Force Pencil.
Resolving Ray.
Resultant Force.
In Direction
and
Magnitude.
§3.
Most of the useful applications of graphical methods treat some system of
parallel forces, in which case, the equilibrium polygon has additional proper-
ties of importance which will now be exhibited.
Let the system of parallel forces be that represented in Fig. 2, viz : let
the verticals 2 3 4, etc., between Wi w^ w^ , etc., be the diagrams of the applied
forces, of which the relative magnitudes are WiW2 , WzW^ , etc., in the force poly-
gon on the right. The force polygon in this case becomes a straight line
(often called the weight line) and the closing side w^Wi of the force polygon is
in the same straight line with the other sides Wi w^ , etc.
Assume any pole p of the force pencil p - ww , and construct the equi-
librium polygon p-ww or ee^ whose sides are parallel to the rays of the force
pencil, in the manner which has been previously explained. Draw the clos-
ing line kk of the polygon ee through the points k^ and h^ where the first and
last sides of the equilibrium polygon intersect the verticals 1 and 6, which last
are assumed to be the lines of support for the applied forces. Draw the clos-
ing Thy pq parallel to kk; thenj as was before shown, it divides the resultant
y .
328 Eddy, Ow ^A^ Two General Reciprocal Methods in Graphical Statics.
force Wi W5 at q into the two parts which rest on the supports in the verticals
1 and 6. The diagram of the resultant is in the vertical mm through the
intersection of the first and last sides of ee^ as was also previously shown.
Choose a second pole p' from which to draw a force pencil p' - mw. Since
this pole y has been taken on a horizontal through j, the new closing line hh of
the equilibrium polygon ccj whose sides are parallel to the rays of this pencil,
will then be also. horizontal. The first and last side will intersect on the ver-
tical mm before found ; and corresponding sides and diagonals of the polygons
ee and cc all intersect in one and the same straight liney^, which is parallel
to^y, as was previously proven. The coincidences just mentioned would, in
any practical case, afford a most comi)lete series of checks and tests of accu-
racy in drawing.
The line pp' and its parallel yy have, in this figure, been made vertical, so
that p and p' are equidistant from ww. Designate the horizontal distance from
p or y to the weight line ww by the letter II. It happens in Fig. 2 that
pwi = //, but in any case the pole distance H is the horizontal component of
the force pq acting along the closing line kk.
Now by similarity of triangles,
kiCi {=z hji^ : ^'2^2 ' ' P^i • 2'tt^i -'^ H. hiC^ = q!W\ . AA = M2 ,
the moment of flexure, or bending moment, at the vertical 2, which would be
caused in a simple straight beam or girder sustaining the given weights WjWj >
etc., and resting upon supports in the verticals 1 and 6.
Again, from similarity of triangles,
A:,/3 ( = A A) : ^a/3 : : // : qy^x
e^fz ( = hji^ : e^s :: H: W1W2
.'. H (^3/3 — e^J's) = H. ^3^3 = qwi . hji^ — w{W2 .hfi^ = Mz^
the moment of flexure of the simple girder at the vertical 3.
Similarly, it can be shown in general that
H.ke = M,
i. e. that the moment of flexure at any vertical whatever (be it one of the
verticals 2 3 4, etc., or not) is the product of the assumed pole distance J7and
the vertical ordinate ke included between the equilibrium polygon ee and its
closing line kk.
Evidently the same properties can be shown to hold respecting the ver-
tical ordinates of the polygon cc, from which it is seen that Arj^a = ^27 etc.,
and H. ke =: Il.hc =, M.
Eddy, On the Two General Reciprocal Methods in Graphical Statics. 329
From the foregoing it appears that the equilibrium polygon for parallel
forces is a moment curve, i. e., its vertical ordinate at any point of the span
is proportional to the bending moment at that point in a girder sustaining
the given weights, and supported by resting without constraint upon piers at
its extremities.
From this demonstration it is clear that
are respectively the moments of the force W1W2 about the vertical 3, about the
vertical mm through the center of gravity, and about the vertical 6.
Similarly, m^^w^ is proportional to the moment of all the forces at the right,
and m^ms to all at the left of the center of gravity, but m,m3 + m^m^ zz , as
should be the case at the center of gravity, about which the moments of the
applied forces vanish.
From these considerations, it appears that the segments inm or nn of the
resultant are proportional to the bending niioments caused by the weights at
the center of gravity of a girder sustaining the given weights and resting,
without constraint, upon a single support at their center of gravity.
Also, the segments y^yz ? ^2^3 ? etc., are proportional to the bending
moments caused at the vertical 6 by the weights W1W2, w^w^^ etc., at the verti-
cal 6, in a girder which sustains these weights, in case it is firmly fixed and
built in at this vertical, and has no other support.
A vertical to which the resultant force is transferred, as it is in this case
to yy, by aid of a couple introduced at that vertical, may be conveniently
designated as a pseudo-resultant. The magnitude of the moment of the couple
here introduced is H .yji^ zny^m^.WiW^^ and by it the closing line kk is often
said to be moved to the position k^y^^ but it seems preferable to call this last a
pseudo-closing line due to the kind of constraint and support at the vertical 6.
§4.
In Fig. 3, let aJ, hc^ cd^ de be the diagrams of the system of applied forces,
and abcde the corresponding force polygon ; choose one point arbitrarily upon
the line of action of each of these forces, and join these points to any assumed
vertex t/ by rays of the frame pencil aHlddd. Also, join successively the
points chosen by the lines hV^ cd^ dd\ which form sides of what may be called
the fram£ polygon.
83
330 Eddy, On the Two General Beciprocal Methods in Graphical Statics.
Now, consider the given forces to be sustained by the frame pencil and
frame polygon as a system of bracing, which system exerts a force at the ver-
tex vf in some direction not yet known, and also exerts a force along some arbi-
trarily assumed member e^ which may be regarded as forming a part of the
frame polygon. From the points bcde in the force polygon, draw the force
lines bb'j c&j dctj e^ parallel to the sides 56', etc., of the frame polygon, and
commencing at a, draw the equilibrating force polygon alfddi^^ whose sides are
parallel to the successive rays of the frame pencil aUfcfdef: then the stresses
upon the rays of the frame pencil will be given in relative magnitude by the
lengths of the corresponding sides of the equilibrating polygon, and the
stresses upon the sides bb% etc., of the frame polygon by the lengths of the
force lines 66', etc. These statements are shown to hold true from the fact that,
in the right hand figure, in which lengths of lines represent magnitudes of
forces, a closed polygon will be found whose sides are parallel to the direc-
tions of the forces meeting at each joint of the frame. The notation, as pre-
viously employed, will assist in the ready identification of the corresponding
parts of the figures on the right and left.
If a resultant side a€ be drawn in the equilibrating polygon, and also
parallel to it from t;', a resultant ray a!ef^ then the ray a!€f will intersect the side
eef at a point in line of action of the resultant of the given system of forces ;
for this intersection is such a point that if the resultant alone be applied at it,
there will be the same stresses along the members a!ef and eef as the applied
forces themselves produce in those members, as appears when we consider that
the triangle aed represents the forces in equilibrium at the point in question.
The diagram of the resultant is ae^ parallel to the side a^ of the force polygon.
If the arbitrarily assumed member eef be revolved about its intersection
with de^ then the force line eef will revolve about ^, and ef will move along the
side rf'^, and it appears that the locus of the intersection of the corresponding
positions of the resultant ray a!d and the last side eef will be the diagram of
the resultant a£.
A first side axx! of the frame polygon can now be drawn corresponding to
the assumed last side ed^ and its significance can be readily seen as follows :
draw the force line aaf^ parallel to this side a/xf-y then the equilibrating polygon
ah'cfdefj the first side of which passes through a, can be begun at any point
of a«'i, and the closing will be parallel to its present position (as is obvious
from mechanical considerations), but the stresses in each of the members of
Eddy, On the Two General Reciprocal Methods in Graphical Statics. 331
the frame will have been changed by this supposition. The case previously
assumed was that in which the stress. in the side oo' is zero.
It is usually a most convenient practical simplification to make all the
sides of the frame polygon lie in one and the same straight line, which may
be called ihe frame line; then, since the force lines are all parallel to it, the
direction of the line ao'i is known at once, (it being one of these parallels), and
the stresses in the rays of the frame pencil are the same, whatever be the
point of aa!i at which the equilibrating polygon is begun.
It should be noticed that the equilibrium polygon is also one case of the
frame polygon.
Suppose the points 8 and 9 of the first and last sides respectively to
become fixed points of support, the thrust between these points may be sus-
tained by a member 8 9, or by 8 and 9 regarded as abutments. To find the
point J of the resultant force ae^ at which it is divided into the two -parts
sustaiited at 8 and 9, draw aQ parallel to t/8, and efQ parallel to ^9, and
then through 6 draw the resolving line q^ parallel to 8 9.
This may be regarded as the same geometric proposition as that pre-
viously proven, in which it was shown that the locus of the intersection of
the first and last sides of an equilibrium polygon (reciprocal to a given force
pencil) is parallel to the resultant side of the force polygon, and is the
diagram of the resultant. The proposition now is that the locus of tjje equi-
librating polygon (reciprocal to a given frame pencil) is parallel to the closing
side 8 9 of the frame polygon and is the resolving line. These two statements
are geometrically equivalent.
Assume a second vertex if, and draw the frame pencil and its correspond-
ing equilibrating polygon ed**(f}f(f. The last point of the equilibrating poly-
gon is at (f or at «'i according as aaC or aaC be taken as the arbitrary side of
the frame polygon.
The intersection of the resulting ray a^^, parallel to the side oTe^ with the
arbitrary side aal* is on the diagram a^ of the resultant, as has been shown.
Also, \i ad be taken as the arbitrary side, it has been shown that o'^S and ^5,
respectively parallel to t/'S and v"9, intersect upon the resolving line jj'.
Again, the corresponding sides of these two equilibrating polygons inter-
sect at 1 2 3 4, points which are upon one and the same straight line parallel
to t/i/"; for this is the same proposition respecting two vertices and two equili-
brating polygons which was previously proved respecting two poles and two
equilibrium polygons.
332 Eddy, On the Two General Beciprocal Methods in Graphical Statics.
For convenient reference, a table of the special names given to the differ-
ent parts of the reciprocal figures just treated is here inserted:
c
a
o
C3
o
2 o
5 ^
Force Diagram,
Frame Pencil,
Frame Polygon,
Resqltant Ray,
Closing Side,
abcde,
afl/&def,
bl/j c&, dd\ ed^
a'€f,
8 9 II q^,
w Diagram of Resultant, ae^
Force Polygon.
Equilibrating Polygon.
Force Lines.
Resultant Side.
Resolving Line.
Resultant Force.
CD
o
D
a
§5.
Let the same system of parallel forces, which was treated by the equili-
brium polygon method in Fig. 2, be also treated by the firanie pencil method
in Fig. 4. Suppose them to be applied to a horizontal girder at 2 3 4 5, and
let it be supported at 1 and 6.
Use 1 6 as the frame line, and choose any vertex r, arbitrarily, from which
to draw the rays of the frame pencil dd. As has been previously shown, if a
resultant ray vo of the frame pencil dd be drawn from v parallel to the closing
line uu of the equilibrating polj^gon rfrf, this ray intersects 1 6 at the point o,
at which the diagram of the resultant intersects 1 6.
Furthermore, the lines w^r^ and d^r^^ respectively parallel to the abutment
rays v\ and vQ of the frame pencil, intersect on the resolving line rr^ which
determines the point of division q of the reactions of the supports, as was
before shown.
Let the vertical distance of the vertex v from the frame line 1 6 be
denoted by the letter V. It happens in Fig. 4 that vQ =, V. When, how-
ever, the frame polygon is not straight, or, being straight, is inclined to the
horizon, Fhas different values at the different joints of the frame polygon ;
but in every case V is the vertical distance of the joint under consideration
above or below the vertex. This possible variation of V is found to be of
practical use in certain constructions.
By similarity of triangles we have
1 2 : t'6 : : r^Vz : w^q .:. V. r^r^ = Wiq . 1 2 zz Jifs ,
the bending moment of girder at 2.
Draw a line through w^ parallel to vS, this line by chance coincides so
nearly with w^Si that we will consider that it is the line required, though it
was drawn for another purpose.
334 Eddy, On the Two General Reciprocal Methods in Graphical Statics.
about that point, of those forces which are found between that abscissa and
the end of the weight line from which this pseudo side was drawn. The dif-
ference between two successive sum totals being the moment of a single force,
a parallel to the pseudo side determines at once the moment of any force
about the point ; e. g. draw dj! parallel to ww, therefore F. rfji' is the moment
of W4W5 about 6.
Now move the vertex to a new position xf in the same vertical with :
this will cause the closing side of the equilibrating polygon (parallel to xfo)
to coincide with the weight line. The new equilibrating polygon hh has its
sides parallel to the rays of the frame pencil whose vertex is at v\ If V is
unchanged, the abscissas and segments of the resolving line are unchanged,
and vvf is horizontal. Also xx parallel to in/ contains the intersections of cor-
responding sides and diagonals of the equilibrating polygon. These state-
ments are geometrically equivalent to those made and proved in connection
with the equilibrium polygon and force pencil.
§6.
In Figs. 2 and 4 we have taken HzzV^ hence the following equalities
will be found to exist :
77ii77i2 = t^^2 , vfiiifn^ = li^d^ , niitn^ := 1^4(^4 , etc.
y,y^ = wA , ffxifz = wA , ViVa = ^4^4 , etc.
m^m^ = d^i , etc. yjc^ = d^i' , etc.
By the^ use of efc., we refer to the more general case of in any applied
forces as well as to the remaining like equalities in Figs. 2 and 4.
From these equations the nature of the relationship existing between the
method of the equilibrium polygon and its force pencil and the method of the
frame pencil and its equilibrating polygon becomes clear. It may be stated
in words as follows :
The hfeight of the vertex (a vertical distance) and the pole distance (a
horizontal force) stand as the type of the reciprocity or correspondence to be
found between the various parts of the figures. The ordinates of the equili-
brium polygon (vertical distances) correspond to the segments of the resolv-
ing line (horizontal forces), each of these being proportional to the bending
moments of a simple girder sustaining the given weights, and resting without
constraint upon supports at its two extremities.
DEMONSTRATION OF A FUNDAMENTAL THEOREM OBTAINED
BY MR. SYLVESTER.
By B. Lipschitz, Professor Ordinarius in the University of Bonn.
In a memoir published in the 85th volume of Mr. Borchart's Journal,
Mr. Sylvester has developed the thought of a new notation for algebraical
forms or quantics. Being given any algebraical homogeneous function of
the n variables o^i, 0^2, . . . ar„, whose degree is denoted by jp, let the powers
and products of powers of the said degree x{^ x^i'~'^X2 - . • ocl be expressed by
-STi, X2, . . . X^, and the corresponding polynomial coeflScients by Tti, Ttj, . . . 7t„,
then the proposed form may be expressed by applying as numerical factors
to the constants /l,/2) • • -f the square roots of the respective polynomial
coeflScients, so that
(1) F {x^, X2, . . . Xn) = VTtT/i-Ti + ^n^f^X^ + . . . + sfnJ.X,,
and is called in that shape a prepared form. Now suppose that on intro-
ducing instead of the n variables oti, 0^2, ... ^„ the n linear functions of n
new variables ^i, ^2? • • • y«j such that
(2) Xa = k^xyx + k^<iy2 + . . . + *fl.n3'n,
where the letter a runs through the numbers 1, 2, ... n, the form F (o^i, x<i^ . . . x^
is changed into the form Gr {y^^y^^. . .y^^ written likewise as a prepared form,
so that
(3) Gr {yu ^2, . . . Vn) = ^nxg^Y^ + ^n^g^Y^ + . . . + ^n^g^Y,.
It is always understood that in substituting for the letter x the letters y, z, ^, w.,
the functions X. are respectively turned into the functions y., Z., T., f7.,
the letter a going through the numbers 1, 2, ... r. As the constant elements
9\i 9i^ * ' • 9^ depend in a linear manner upon the elements /i,/2, ♦ . ./^, we
get by partial diflferentiation tlje v equations
(4) 9i = -ij J\ + ~^j /2 + • • • + -TF f^ y
^ — ^9- f i ^9^ ft i ^9y f
836
338 LiPSCHiTZ, Demonstration of a Fundamental Theorem^ &c.
that contains the constants Ai , A2 , . . . A^ . Now we may observe that, on solving
the system (8) with respect to the quantities z^ , there arise the equations
(10) 5J5 := ki^ ifXi + ^2^ 5^2 + • • • + ^n, b'^^n >
where the system of coeflScients is got from (5) by interchanging the hori-
zontal and vertical lines with each other, or by transposition. Consequently,
the form H{zij Z2j . . . z^) is transformed into the form -F (org , ^2 > • • > ^») by
the substitution
(11) ^1, 1 > ^2, 1 > • • • • ^n, 1 >
which is the substitution (5) transposed ; wherefore, if we form the equations
^'-M.^'^fi^'^' '•• + m[*^'
it is evident that the coefficient ^ may be formed from the coefficient ^* by
changing the substitution (5) into (11), or by changing k^i, into kj,,,. Con-
sidering the eflfect of the equations (8) upon the form F {xi^ X2j . . . x^ we
must say that the substitution (7) operated on the variables induces the sub- '
stitution operated on the elements
/-ION hhi hhi hlik
hhy Shy Shy
Therefore, it is to be shown, that the substitution (13) is contrary to the sub-
stitution (6) . But a general property of partial diflferential coefficients of n
functions taken according to n independent variables teaches us that the sub-
stitution contrary to (13) is represented with the aid of the partial diflferential
coefficients of the n variables taken according to the n respective functions, as
follows :
LiPSCHiTZ, Demonstration of a Fundamental Theorem^ &c. 339
(14) ^\ ^% . . . . ^^ ,
hhi hhi Ski
Sky Sky Sky
Of course our theorem only requires us to establish that the substitu-
tions (6) and (14) accord with each other, or that for any combination of the
numbers a, (i the equation
^ ^ . Sk, Sf^
is valid. Meanwhile, we have seen that the partial diflferential coefficient ^? is
changed into ^ by changing k^^ into Ar^a. Consequently, it will suffice to
ok^
prove, tkat tke partial differential coefficient ^ turns into tke partial differential
coefficient ^ , if k^^b i^ changed- into kf, „ .
Supposing that the forms in question are expressed in the usual manner,
(16) J?* (^, , 0^2 , . . . . ^n) = Ttii^i^i + 7122^2^2 + • • • • TtyFyXy ,
(17) Gr (yij ^2} • • • • y«) = ^1^1 ^1 + '^Cr2^2 + . • • . TlyGyYy ,
(18) -ff (5^1 , 5^2 5 • • • • ^n) ^^ TtiHiZi -f- 7I2H2Z2 + . . . . TtyllyZy ,
the constants /., ^^ , ky are connected with the constants i^., G^^, ff^j by the
purely numerical relations,
(19) /. = Vn^F, , ffp = VnpGp , k^ = ^Jn^Ey ,
so that, instead of (15), we have the equation
/ork\ ^^« . SGa
In order to prove the former, we are going to prove the latter. Taking notice of
SG- ' SF
the fact that the partial differential coefficient z~ turns into ytv ^Y changing
k^ I, into kf,^ „ , the meaning of (20) may be expressed in the words, tkat tfie
product Tta j-^ is ckanged into tke product n^ ^-- by ckanging k^ b i^to k^, ^ .
obf^ SF^
As it is permitted to regard the quantities Y^ as linear functions of the
quantities X^ and reciprocally, by differentiating the equation
F (xi, X2, . . . Xn) = (} (yi, 3^2, .. . Vn) '
first according to F^ , afterwards according to y^ , we shall find
340 LiPSCHiTZ, Demonstration of a Fundamental Theorem^ &c.
In like manner from the equation F (x^ X2j . . . ar„) zn H {zi^Z2j . . . z^) result
the equations
(22) 2:,'"'^,'-'''^'' ""^ m,- ""' ^xv
Whence it is evident, that the equation (20) will be proved true if the equation
(23) 7t. Y^F — ^,
hY,-'''hX.
is proved to hold good.
This equation containing no trace of the respective forms, let us denote
hy ^, ^, . . . ^» a set of n new independent variables, with which we form the
expression
(24) AXi + ^^2 + . . . + *n^n J
which is to be elevated to the pth power. By the aid of the equation (2) and
of the definition
(25) Uj, = Jc^^^ty + Jc^^J^ + . . . + Kbtn
the expression (24) assumes the following shape
(26) u^yi + ^23^2 + . . . + «^«y«-
Hence arise by means of the previously introduced notation the expressions
(27) {t,x, + t,x, + ... t^x:)^=n,T,X, + n,T,X, + . . . + Tt^T.X,
(28) («^i3^i + t^y2 + . . ' + u^yny = nrU,Y^ + 7t^U2Y2+ . . . + n,U,Y,.
Considering that X. and Yp as well as T^ and Up depend linearly on one
another, we may diflferentiate the equivalent expressions (27) and (28), first
according to T., afterwards to Y^, and get
SUa .. 5X hU
(29) 7r.X.= ^^^i^jf^f^'^ ^
SY-""' ST'
But as by virtue of (25) and (10) the variables Uj, depend upon the variables
ta in the same way as the variables Zi, depend upon the variables ar«, we con-
clude that the quantities U^ must be the same linear functions of the quan-
tities T. as the quantities Z^ are of the quantities X^j and that consequently
the partial diflferential coeflScients ^—^ and y—? denote the Same thing. Hence
oT^ dX^
it follows that the second equation in (29) produces the equation (23) as was
to be proved, and thus the demonstration of the proposed theorem is accom-
plished.
342 Sylvester, Note on the Tlieorem Contained in Prof, LipscUtz's Paper.
sions which flow from this conception, we have only to add the rule that all
combinations of invariants, or of covariants, or of contravariants, etc., are them-
selves invariants, or covariants, or contravariants, et^., respectively. We may
then state that the effect of substituting in a covariant or contravariant (to a
prepared form) in place of the variables, or in place of the coeflScients, the
symbolic inverses of the one or the other, is to reverse their character and con-
vert covariants into contravariants and vice versa, leaving of course the charac-
ter of invariants unaltered ; and I may remark incidentally that we are thus
provided with a means of making any two invariants operate on each other
so as to produce a third, a mode of operation which was not possible previous
to the introduction of the prepared form.*
Moreover, the word combination must be taken in its widest sense, as
there is more than one mode of combination possible. For example, if F, (?, II
are covariantive and <I> a contravariantive function of .(a, 6, c, . . ; or, y, 5J, . . . ),
where a, b^ c^ . . . are the coeflicients and x,y, z, . . . the variables of a
prepared form, we have of course 4> f — j;??^---; ^ ^V ^ - -jP ^^d
of them covariants. This may be termed an external mode of combination,
but we shall equally have covariants derived by an internal mode of combi-
,(dF dP \ ^( J. dP dP \
nation, ex. gr. 4)^— , ^ » • • • ; ^ » y > • • J, ^y^y o , . . .\ — , _ , . . J,
^fdP dP da dG \ ... . .
Hirr-, -iT- J • • • ; -T- > ^- ? • • • ] Will also be covariants.
\da db dx dy /
So again it may be observed that these modes of combination admit
of being applied in more than one Vay : thus, to confine ourselves for a
moment to the case of two -forms, their external operation on each other
may be simple, or concurrent, or reciprocal: simple when in one of them
one set of quantities are converted into operators, concurrent when both sets
are so converted, but reciprocal when in one of the two forms the variables
land in the other the coefficients undergo such conversion. As an example,
* The ca«e may be stated thus : previous to the introduction of the prepared form^ invariants of systems
could be made to operate upon invariants solely through the instrumentality of. the coefficients of the linear
forms of the system ; since its introduction the same operation may be made to take effect through the instru-
mentality of the coefficients of all the forms, linear or nun-linear, indiscriminately. The first named mode of
operation is equivalent to the hyperdeterminantive method, which includes that of Ueherschiebung ; the latter
transcends the sphere of byperdeterminants.
Sylvester, Note on the Theorem Contained in Prof. Lipschitz^s Paper. 343
suppose we take the prepared form aaf^ + . . . -\- d^ , and its skew covariant
{c^d + . . . ) or* 4- • • • — i^d^ + • • • ) y^- ^^ ^^y combine the contravariant
i-j- •^ + • • • + ;r7 y^) with the contravariant (a^d + . . . ) (-r-j + • • •
3
— (ad!^ + • • • ) ( T") ? ^^d th^ result will be a numerical multiple of the
contravariant d^ + . . . — «/. If, in the above instance, we denote the square
of the primitive and its skew covariant according to their degree and order by
2*6, 3 • 3 respectively, we may explain their mutual action stenographically by
saying that 2 • 6 and 3 • 3 have acted reciprocally on each other, the dot signi-
fying that the quantities typified by the number so marked have been replaced
by their symbolic inverses ; we cannot well represent this mutual action by
2-6
writmg 2*6'^3"3or3'3'^2*6, but may employ for the purpose * . .
So from the square of a quintic 2 • 10 and its linear covariant 5 • 1 we may
2 • 10
derive by reciprocal action * . , or the contravariant 3*9: or, again, we
5* 1
may take any even number of covariants and cause them to operate in various
manners, the variables on the variables and the coefficients on the coefficients,
so as to form a closed circuit, as ex. gr. with four, we may make the coefficients
of the first operate on those of the second, the variables of the second on those
of the third, the coefficients of the third on those of the fourth, and the varia-
bles of the fourth on those of the first. Thus we have passed from reciprocal
to the more general notion of simultaneous or circulatory action between any
even number of covariants. And it is not unlikely that further applications
may be made of this fertile conception : when dealing with a principle (an
intellectual force) as distinguished from a theorem (a mere law), we never can
feel sure that its uses are exhausted, or its plastic power spent.
LETTER FROM MR. MUIR TO PROFESSOR SYLVESTER ON
THE WORD CONTINUANT.
In reference to the foot-note on page 127 of your Journal of Mathe-
raatics, will you allow me the use of a corner to state that the name
" Continuant^^^ such as it is; was chosen by me before I had had the pleasure
of profiting by your papers in the. Philosophical Magazine, and when I
believed myself to be the first who had lit upon the peculiar determinant-
form in question. On being undeceived by finding the discovery attributed
in several German writings (see especially Gunther's Darstellung der Ndherungs-
werthe von Kettenhruchen in independenter Form) to continental mathematicians,
I carefully examined into the matter, testing the validity of the claims made
in these writings, and seeking for earlier traces of the discovery. The result
of this work convinced me of your undeniable priority, and when next I had
occasion to write on the subject (Phil. Miag., 1877) I called attention, in the
most pointed manner, to the conclusion I had come to, and gave the necessary
reference to your papers. I have not, however, ceased to use the term " con-
tiniuint,^^ (1) because, as an exceedingly suitable and euphonious abbrevia-
tion for *• contimted'Jraction determinantj^^ it seems to me to be the very word
wanted, (2) because, in this way, it is a short literal translation of the equiva-
lent term ^^ Kettenhruch-Determinante^^^ which is the received name in Grermany,
(3) because, though it may be somewhat scant of meaning to a literalist, I
cannot but consider it eminently " suggestive," and (4) because doubtless I
have still a foster-father's kindly feeling towards the name he has known
another's child by.
4th Seftembkb, 1878.
[Reasons 2 and 3 above given appear to aflford quite a suflScient justifica-
tion for the use of the word in question. J. J. S.]
844
EXTRACT FROM A LETTER OF DR. FRANKLAND TO
MR. SYLVESTER.
... I WANT now to oflfer one or two critical remarks upon your papers. . . .
At page 65 you notice the anomaly of d^jree bond. My system of notation repu-
diates all free bonds ; but I fear the expression may sometimes have escaped
me in describing the transference of an element or compound radical from
one compound to another ; but I only meant the breaking of the bond for an
infinitely short time. I think you also employ the corresponding term ** free
valence" in this sense at page 69. For free valence, however, I should sub-
stitute "latent valence." With very few exceptions, this latent valence is
always an even number, hence my hypothesis of two bonds saturating each
other (see page 21 of my Lecture Notes).
At page 71, I see you prefer Ladenburg's prism or hexagon to Kekule's
graph for benzol.* It certainly accords better with facts. The facts are
1st. When 1, 5 or 6 atoms of hydrogen in benzol are replaced by (7/, ifr,
&c., there are no isomers. 2d. When 2, 3 or 4 atoms are so replaced, there
are always 3 isomers and three only. Now, Kekule's graph exhibits
the possibility of four different positions for 2, 3 or 4 atoms of C/, and hence
of 4 isomers.
ci
ci
Cl
a
H
/
H
Cl <\
H
H H
No. 4.
Kekule does not seem to have noticed the diflference between No. 1 and
No. 4.
* I did not intend to express any such preference, but merely to state the mathematical fact that the graph
of the one is a graph of a covariant belonging to a single biquadratic form which the other is not. The truth
is, that chemical graphs correspond to the case of. an indefinite repetition of each algebraical form as would
necessarily be the case if each chemical atom were capable of playing a different part, or, at all events, of being
distinguished in character from every other of the same name with which it may be associated. Mr. Lockyer's
"ecent discoveries seem to favor the notion of this being the case for hydrogen. J. J. S.
87 345
346
Extract from a Letter of Dr. Frankland to Mr. Sylvester.
mo. e.
On the other hand, the graph, No. 5, will not answer, because it only
allows of two. different positions, AA and J5J5, unless indeed you assign a
diflferent value to the cross bonds from that of the lateral ones ; but No 6 gives
three different positions, and three only, viz : 1st. Either 1-2, 2-3, 3-4, 4-5, 6-6,
or 6-1, for all these pairs are obviously identical. 2d. Either 1-3, 3-5, 6-1, 2-4,
4-6, or 6-2, for these positions again are identical ; and 3d. Either 1-4, 2-5,
or 3-6, which are also identical. We are now accustomed to call the first set
of positions " Ortho," the second " Meta" and the third " Para." I quite
agree with you that all graphs may be represented by figures in a plane.
Thus, No. 6 may be drawn as No. 7. It is very amusing to see the figure
worship of some chemists, and the horror of it in others. Both seem to be
unaware that graphic formulae are only symbols of phenomena, and have no
connection whatever with space
I was at first inclined to smile at your invention of a new set of atoms
(atomicules); but, on further consideration, it seems to me that your concep-
tion of the constitution of atoms may prove of great value, not in its applica-
tion to variation of atomicity, but in furnishing an explanation of the beha-
viour of certain atoms which has long puzzled me and doubtless also other
chemists. Carbon aflfords the best example of what I mean. The strong
aflSnity of carbon for carbon is a quality upon which depends the very exist-
ence of nearly every organic compound, and sharply distinguishes these
compounds from the almost equally complex siliceous minerals, in which the
atoms of silicon are, in no single instance, combined with each other. The
most obvious explanation would be that the bonds of carbon are + and — to
each other, whilst those of silicon are identical or endowed with energy of
the same kind and intensity. But it is a remarkable fact that hitherto, with
very few and dubious exceptions, aH attempts to establish a diflFerence between
the 4 bonds of the carbon atom have failed. Thus, if an atom of carbon be
I +
combined with 4 atoms or monad groups a, ft, c, df, -c-C-a of which a and b
I
^d
Extract from a Letter of Dr. Frankland to Mr. Sylvester. 347
are + and c and d — , it does not matter apparently in what order these
+
bodies are introduced to the carbon atom ; for instance, if the body a be
. — — +
replaced by a negative body e^ and c by a + body /; and if then in another
molecule of -c-C-a^ c be replaced by e^ and a by/, the two new molecules
-d
are identical and not isomeric as might be expected. But if, according to
your conception, each bond is tripartite, it could of course exert either + or —
energy according to the quality of the atom or group presented to it.
The application of the theory of atomicules to explain latent bonds is of
course perfectly legitimate, but I cannot see its advantage over the hypothesis
of mutual neutralization. It has never occurred to me that there is any
more difficulty in thinking of carbonic oxide as CC = than in conceiving
methyl as Q^nff . If two bonds of two separate but perfectly similar atoms
of carbon can satisfy each other, why not two bonds in one and the same
atom ? If you could apply your theory of atomicules to explain the small
number of anomalies which have hitherto defied all other theories, chemistry
would be greatly indebted to you. Thus, nitric oxide, if written ^ = 0, as
required by Avogadro's law, violates the law of variation of atomicity, for a
perissad element becomes an artiad. If, on the other hand, it be written, as
CN=0
in my notes, i it violates Avogadro's law for equal volumes of hydro-
gen and nitric oxide do not contain an equal number of molecules.
In reference to your remark on p. 83, every chemico-graph is assumed to
be possible, but, in many cases, it is known to be incapable of existing under
the conditions prevailing at the surface of the earth.
At page 108, you seem to arrive, by a mathematical process, at
H-C^O-H
11 as the graphic formula of aldehj^de ; but this is the formula
H " C— H
assigned by chemists to vinylic alcohol, which appears, however, to be
incapable of existence. The formula universally recognized for aldehyde
H H
is H-C" H which makes its relations to acetic acid H-C-H and
348 Extract front a Letter of Dr. Frankland to Mr. Sylvester.
H
alcohol TT ^n r^ TT simpler than yours.*
r
//
The law you mention at page 114, "m n-^alent atoms may he replaced
by n ni-valent ones,^^ has a very wide application in chemistry.
I do not think that we have any trustworthy evidence of the difference of
5" atoms mentioned at page 123.
I cannot, like Mr. Mallet, conceive of "centres of force," or indeed of
force at all, apart from matter.
In reference to the origin of the theory of atomicity, I send you, by book
post, a copy of a paper of mine presented to the Royal Society, on May 10th
1852, in which, at page 438, et seq.^ you will find what, so far as I know, was
the first enunciation of the theory of atomicity. After some preliminary
explanations the law is expressed as follows, at page 440 : " Without offering
any hypothesis regarding the cause of this symmetrical grouping of atoms, it
is sufiiciently evident, fi'om the examples just given, that such a tendency or
law prevails, and that, no matter what the character of the uniting atoms may be,
the combining power of the attracting element, if I may be allowed the term, is
always satisfied by the same number of these atoms.^^ I then go on to apply the
theory to the explanation of the constitution of organo-metallic compounds.
It was in fact the discovery of these bodies which first forced the law upon
my attention ; for I was surprised to find that the more methyl or ethyl I
combined with a metal, the less room was there for 0, CI, &c., although the
energy with which the ethylised metal combined with these elements was
enormously greater than before ethylisation, some of the ethylised metals
being even spontaneously inflammable. If you remember that what I call
" the uniting atoms " were monads, and that nearly all chemists then regarded
the atom of oxygen (with the atomic weight of 8) as equivalent to one atom of
hydrogen, you will readily understand the examples I give. The application
of the theory to carbon compounds generally, was made by Kolbe and myself
in December, 1856 (see Ann. der Chemie. und Pharm., Bd. CI, s. 257), the
first word from Kekule on the subject of atomicity appearing about a year
* If my memory is not in fault, the form 1 have given is the only graph possible if each carbon atom refers
to the same biquadratic, but there seems no reason to suppose that such is the case in nature. {VifU fout-note,
page 845.) J. J. 8.
Extract from a Letter of Dr. Frankland to Mr. Sylvester. 349
later. I have ordered my publisher to send a copy of my "Experimental
Researches " to the Library of the Johns Hopkins University. If you will
kindly refer to pages 145 and 154, you will see what I have said on the sub-
ject last year, whilst at page 188 you will find the paragraphs in my Royal
Society memoir, and at page 148 the manifesto of Kolbe and myself which
discloses the application of the doctrine of atomicity, or atom fixing power,
to the compounds of C. Look especially at page 150 for the declaration of
Kolbe's conversion to the theory. All this happened, as I have said, long
before Kekule took up the matter ; but his subsequent work, and especially
the later rectification of atomic weights by Oannizzaro, aided the further
development of the theory enormously ; but it is singular that no subsequent
writer on the subject appears to have known, of my paper in the Philosophical
Transactions.
I trust that you will go on with the consideration of chemical phenomena
from a mathematical point of view, for I am convinced that the future pro-
gress of chemistry, as an exact science, depends very much indeed upon the
alliance of mathematics.
London, October IS^ 1878.
88
APPLICATIONS OF GRASSMANN'S EXTENSIVE ALGEBRA.
By Professor Clifford, University College^ London.
I PROPOSE to communicate in a brief ^orm some applications of Grass-
mann's theory which it seems unlikely that I shall find time to set forth
at proper length, though I have waited long for it. Until recently I was
unacquainted with the Ausdehnungslehre, and knew only so much of it
as is contained in the author's geometrical papers in Crelle^s Journal and
in HankeVs Lectures on Complex Numbers. I may, perhaps, therefore be
permitted to express my profound admiration of that extraordinary work,
and my conviction that its principles will exercise a vasl influence upon the
future of mathematical science.
The present communication endeavors to determine the place of Qua-
ternions and of what I have elsewhere* called Biquaternions in the more
extended system, thereby explaining the laws of those algebras in terms of
simpler laws. It contains, next, a generalization of them, applicable to any
number of dimensions ; and a demonstration that the algebra thus obtained
is always a compound of quaternion algebras which do not interfere with
one another.
On the Relation of Grassmann^s Method to Quaternions and Biquaternions; and on
the Generalization of these Systems.
Following a suggestion of Professor Sylvester, I call that kind of multipli-
cation in which the sign of the product is reversed by an interchange of two
adjacent factors, polar multiplication; because the product ah has opposite
properties at its two ends, so that ah = — ha. The ordinary or commutative
multiplication I shall call scalar^ being that which holds good of scalar num-
bers. These words answer to Grassmann's outer and inner multiplication ;
which names, however, do not describe the multiplication itself, but rather
those geometrical circumstances to which it applies.
* Proceedings of the London Mathematical Society.
850
Clifford, Applications of Grassniann's Extensive Algebra. 351
Consider now a system of n units <i, (2? • • • u^ such that the multiplication
of any two of them is polar ; that is, i^ i, = — i^i^. For geometrical applica-
tions we may take these to represent points lying in a flat space of n — 1
dimensions. A binary product i^i, is then a unit length measured on the line
joining the points t^ , t, ; a terna.ry product ir i, it is a unit area measured on the
plane through the three points, and so on. A linear combination of these
units, 2«Aj = oc suppose, represents a point in the given flat space of n — 1
dimensions, according to.the principles of the barycentric calculus, as extended
in the Ausdehnungslehre of 1844.
In space of three dimensions we may take the four points (q? «n h^ h so
that ti, L2J tz are at an infinite distance from iq in three directions at right
angles to one another.
Now there are two sides to the notion of a product. When we say
2X3 = 6, we m^y regard the product 6 as a number derived from the
numbers 2 and 3 by a process in which they play similar parts ; or we may
regard it as derived from the number 3 by the operation of doubling. In the
former view 2 and 3 are both numbers ; in the latter view 3 is a number,
but 2 is an operation, and the two factors play very distinct parts. The
Ausdehnungslehre is founded on tlie first view ; the theory of quaternions on the
second. When a line is regarded as the product of two points, or a parallelo-
gram as the product of its sides, the two factors are things of the same kind
and play similar parts. But in such a quaternion equation as jp = cr, where p
and a are vectors, the quaternion j is an operation of turning and stretch-
ing which converts p into a ; it is a thing totally different in kind from the
vector p. The only way in which the factors j and p can be taken to be of
the same kind, is to regard p as itself a special case of a quaternion, viz : a
rectangular versor. But in that case the expression does not receive its full
meaning until we suppose a subject on which the operations p and j can be
performed in succession.
The quaternion symbols ?, j, k represent, then, rectangular versors; that is
to say, they are operations which will turn a figure through a right angle in
the three coordinate planes respectively. It follows that if either of them is
applied twice over to the same figure, it will turn it through two right angles,
or reverse it ; we must therefore have t? =if =1 1(? z= — 1 .
To compare these with the symbols for the four points to? 'n h) ^3? let us
suppose that i turns the line 1^12 into i^i^] that j turns iql^ into toii; and that k
852 Clifford, Applications of GrassnianrCs Extensive Algebra.
turns (o^i into io«2- The turning of 1^12 into iqi^ is equivalent to a translation
along the line at infinity /a's- We may, therefore, write i = t2t3, and so
j =z= (3 (, , A: = ti 12 • Now i turns i© h into i^ i^ ; that is
or iq 63 = 12 (3 . to ^2
= — ii • ^0 '3 •
We are therefore obliged to write il = — 1 , and in a similar way we may
find t? = ^3 = — 1.
This at once enables us to find the rules of multiplication of the i, J, k.
Namelv, we have
JtC = I3 Li . Li I2 = I2 i-z = ^
tCl = ii l2 . I2 ^3 ^^^ ^3 '1 ^^^ ^
(? = t2 (3 • ^8 «1 = «1 '2 = ^
and finally
ijk = i2 Is . (3 «i • '1 '2 = — 1 •
In order, therefore, to being the quaternion algebra within that of the
Ausdehnungslehre, we have to make the square of each of our units equal
to — 1, as pointed out by Grassmann.* But I venture to differ from his
authority in thinking that the quaternion symbols do not in the first place
answer to the " Elementargrosse " of the Ausdehnungslehre, but to binary
products of them ; from whioh supposition, as we have seen, the laws of their
multiplication follow at once.
It is quite true that in process of time the conception of a product as
derived from factors of the same kind, and so of the product of two vectors
as a thing which might be thought of without regarding them as rectangular
versors, grew upon Hamilton's mind, and led to the gradual replacement of
the units i,j^ k by the more general selective symbols S and F. Tp explain
the laws of multiplication of i, ^*, k on this view, we must have recourse to the
theory of "Erganzung," or, which comes to the same thing, represent an
area ij by a vector k perpendicular to it. But the explanation in this case is
by no means so easy ; and it is instructive to observe that the distinction
between a quantity and its " Erganzung," i. e. between an area and its repre-
sentative vector, which, for some purposes it is so convenient to ignore, has to
be reintroduced in physics. Thus Maxwell specially distinguishes the two
kinds of vectors which he calls force and flow^ and which in fact are respec-
tively linear functions of the units and of their binary products.
■— — — ■-- ■ - ■ - - - - — — ■ — ■ - ■ ■ _ _ ^
*Math. Annalen.
Clifford, Applications of Grassmann's Extensive Algebra. 363
We have regarded the symbols t, J, k as rectangular versors operating on
the quantities co *i > hh^ h h • These quantities are unit lengths measured any-
where on the axes in the positive directions. They have magnitude, direction,
and position, and are thus what I have called rotors (short for rotators) to dis-
tinguish them from vectors^ which have magnitude and direction but no posi-
tion. A vector is of the nature of the translation- velocity of a rigid body, or
of a couple ; it may be represented by a straight line of given length and
direction drawn anywhere. A rotor is of the nature of the rotation- velocity of
a rigid body, or of a force ; it belongs to a definite axis. A vector may be
represented as the difference of two points of equal weight (the vector ah may
be written 6 — a); this is shewn by the principles of the barycentric calculus to
represent a point of no weight at infinity. Accordingly the symbols ii, cg, «3 may
be taken to mean unit vectors along the axes. In fact, if we write co + ^r = ocr »
the points a will be situated on the axes at unit distance from the origin, and
thus ir=-ar — Co will represent the unit vector from the origin to a^ .
The versors t, j, k will operate on these vectors in the same way as on the
rotors to h ) hh ? *o ^ • We find that ii2 =z 1213* 12^=: izj jtz = ti , kii zz cj • These
rules of multiplication coincide with those for t, j, k if we write the latter in
place of ci, i2j cs- Thus we may use the same symbols to represent unit vectors
along the axes and rectangular versors about them. But it is not in any
sense true that the vectors ti, 12? h are identical with the areas laCa, hiiy iii2]
it is only sometimes convenient to forget the difference between (1 and 12^3.
In the elliptic or hyperbolic geometry* of three dimensions, the four
points LoyVuhyh must be taken as the vertices of a tetrahedron self-conjugate in
regard to the absolute, so that the distance between every two of them is a
quadrant The product of four points a/?yS will then consist of three kinds
of terms ; (1) terms of the fourth order, being i^ c^ 12 £3 multiplied by the deter-
minant of the coordinates of the four points, which is proportional to sin (a, /?)
sin (y,i) cos (a/?,yi); (2) terms of the second order, resulting from pro-
ducts of the form iliih^ = — h h ; (3) terms of order zero, resulting from
products of the form cj , tj ij . Altogether we may arrange a^yh in eight terms
as follows :
a^yh = a + 2ftr* «r «« + <? Co h h h • [^j ^ different.]
And it is now easy to see that the product of any even number of linear fac-
tors will be of the same form. This form is what I have called a hiquaternion^
* Dr Klein's names for the geometry of a space of uniform positiye or negative caryature. See Froc.
Lond. Math. Soc
89
354 Clifford, Applicatio)is of GrrassmanrCs Extensive Algebra.
and may be easily exhibited as such. Namely, let us write cj for iQiii2iz ; then
we have
i = '2 '3 j = hh k = li (2
ui = io = /i to J ^j = j^ = h Iq, o)k = ku = (3 iq
6)^= 1.
Therefore, the product of any even number of factors greater than two is a
linear function of 1, i, j, yt, o), oi, oj, coA; ; that is to say, it is of the form q + car,
where j, r are quaternions. While the multiplication of u with i, j, ^ is scalar,
its multiplication with iq, ti., i2j h is polar. The effect of multiplying by ca is to
change any system into its polar system in regard to the absolute.
The chief classification of geometric algebras is into those of odd and even
dimensions. The geometry of an elliptic space of n dimensions is the same
as the geometry of the points at an infinite distance in a flat or parabolic
space of n + 1 dimensions ; the theory of points and rotors in the former is
the same as that of vectors and their products in the latter. Each requires a
geometric algebra of n + 1 units. Thus the algebra of four units, leading as
above to biquaternions, is either that of points and rotors in an elliptic space
of three dimensions, or of vectors and their products in a flat space of four
dimensions. All geometric algebras having an even number of units are
closely analogous to it ; of these I would point out particularly that of two
units, belonging to the elliptic geometry of one dimension or to the theory of
vectors in a plane. Let the units be (2 > «3 ; then a product of any even
number of linear functions must be of the form a + h^is. Let i= i2isy
theni^=: — 1; and such an even product is the ordinary complex number
a -{- bi. In the method of Gauss every vector in the plane is represented by
means of its ratio to the unit vector (gj that is to say, tg and cg are replaced
by 1 and i. This gives an artificial but highly useful value for the product of
two vectors. We might apply a similar interpretation to the algebra of four
units, denoting the points iq, ti, t2, h t>y the symbols o), i, J, ^, and consequently
their polar planes oiq , oii , ota » (*>^s t>y the symbols 1 , cot , oj , ok ; but I am not
aware that any useful results would follow from this imitation of Gauss's
plane of numbers.
Bules of Multiplication in an Algebra ofn units.
In general, if we consider an algebra of n units, (i, «2j • • • ^o such that
/^ = — 1 , tr = — '«'r J a product of m linear factors will contain terms which
are all of even order if m is even, and all of odd order if m is odd ; for the
* Clifford, Applications of Grassmann^s Extensive Algebra. 356
•
substitution of — 1 for any square factor of a term reduces the order of the
term by 2.
A product of m units, all different, multiplied by any scalar is called a
temi of the order in. The sum of several terms of order m, each multiplied
by a scalar, is a form of order m. The sum of several forms of diflferent
orders is a quantity and an even quantity when the forms are all of even
order, an odd quantity when they are all of odd order. Thus the multipli-
cation of linear functions of the units leads only to even quantities and odd
quantities.
The square of a term of the m^ order 1 5 + 1 or — 1 according as the integer
part of -^{m + 1) is even or odd. For the product «i «2 • • • u «i «2 • • • «m is
transformed into 1J12 • • • d by ^ m {m — 1) changes of consecutive factors, and
therefore equals ± 1 according as — m (m + 1) is even or odd ; which is
equivalent to the rule stated.
The multiplication of a term P of order m by a term Q of order n, having k
factors common^ is scalar or polar according as mn — ^ is even or odd. Let
P = CP' and Q = CQ, where C, P, Q have no common factor; then the
steps from CP'CQ' to CP'QC, CQFC, CQCP' require respectively k{n — k),
(m — k){n — A:), k {m — k) changes of consecutive factors; and the sum of
these quantities is even or odd as mn — Ar^ is.
The following cases are worth noticing :
(1) When two terms have no factor common, their multiplication is
scalar except when they are both of odd order. (Case A = 0).
(2) The multiplication of two even terms is scalar or polar according as
the number of common factors is even or odd.
(3) If one of two terms is a factor in the other, the multiplication is
scalar except when the first is odd and the second even.
Theory of A Igebras with an odd number of units.
When the number of units is n = 2//i + 1 , there are n terms of the order
n — 1 , and all terms of even order can be expressed by means of these. For
the product of any two of these terms is of the second order, since they must
haven — 2 factors common. We obtain in this way all the terms of the
second order; and from them we can build up the terms of the fourth, sixth
356 Clifford, Applications of Grassmann^s Extensive Algebra.
•
orders, etc. Let the product of all the units ti t2 . . • u be called o, then these
terms of the order n — 1 shall be defined by the equations kr= oi^^ It will
follow that kikz . . . k^=:^ I according as m is even or odd, or, which is the
same thing, according as the squares of the k are + 1 or — 1» By means of
this formula, terms of order higher than m in the A, may be replaced by
terms of order not higher than m. The multiplication of the k is always
polar.
The terms of even order, regarded as compound units, constitute an
algebra which is linear in the sense of Professor Peirce, viz: it is such that the
product of any two of these terms is again a term of the system. The num-
ber of them is 2~~^= 2^"*; for the whole number of terms, odd and even,
is l + w+-n.n — l + ... + n + l=(l + ir=2~, and the number of
even terms is clearly equal to the number of odd terms.
I shall call the algebra whose units are the even terms formed with n ele-
mentary units (i (2 • • • ^n > the n-way geometric algebra. Thus quaternions are
the three-way algebra. We may regard the units of quaternions as expressed
in either of two ways. First, in terms of the elementary units iii2h'j they
are then (1 , t^'s » ta ^i , ti i^) . Secondly, we may write Ati, Atj for the terms i^h i h^u
and the system may then be written (1, ki, Atj , k^ k^). In this second form it
is identical with the entire algebra of two elementary units, including both
odd and even terms.
The five- way algebra -depends upon the five terms ^i, A:,, ^s, ^4, Arg and their
products; the number of terms is sixteen. Now we may obtain the whole of
these sixteen terms by nmltiplying the quaternion set
(I J ^1 f ^2 > ^1 ^2)
by this other quaternion set
For each of the sixteen products so obtained is a term of the even five- way
algebra, and the products are all distinct. Moreover, the two quaternion sets
are commutative with one another. For since the k multiply in the polar man-
ner, we may regard them as elementary units for this purpose ; now the terms
in the second set are all even, and no term in one set has a factor common
with any term in the other set.
In the language of Professor Peirce, then, the five-way algebra is a com-
pound of two quaternion algebras, which do not in any way interfere, because
the units of one are commutative in regard to those of the other. A quantity
Clifford, Applications of GrassmanrCs Extensive Algebra.
357
in the five- way algebra is in fact a quaternion o + ix +ji/ + kzj whose coeflS-
cients q x y z are themselves quaternions of another set of units (1 , ?i , ji , A:,) ,
the «i , ji , ki , being commutative with iy j j k .
I shall now extend this proposition, and shew that the {2m + Vj-way alge-
bra is a C09npound of m quaternion algebras^ the units of which are commutative
with one another. To this end let us write po -zzk^k^y and then
P\ m IC\ ICi ATg ICi HI ^0 IC^ Ki ?! — - ^3 ^4 ^5
j>2 = i>i ^10 K\ q2 = qi ks kg
Pr — Pr — 1 ^4r + 2 ^4r + 3
Consider now the quaternion sets
Jr — 2r - 1 ^4r ^4r + 1 •
2* i* §/* L* 1/* 1/*
> A.4 /l5 ) /tj /t3 , 71^3 71^4
J J>1 A^io J Pi kii , Aio A^ii
J Jr — 1 ^4r J ?r— 1 ^4r + 1 ) ^4r ^4r + 1
J Pr — 1 k^r + 2 J Pr — 1 ^4r + 3 » ^4r + 2 ^4r + 8
viz : a2>-set and a j-set alternately. I say that if we consider the first m sets
of this series, we shall find them to involve 2m + 1 of the k\ that the pro-
ducts of m terms, one from each series, constitute 2^*" distinct terms, which are
therefore identical with the terras of the {2m + l)-way algebra; and that the
terms in any two sets are commutative with each other. The first two
remarks are obvious on inspection ; the last also is clear for the case of a ^-set
and a j-set, because the j-set is of even order in the A:, and no factors arc com-
mon to the two sets. It remains only to examine the case of two j>-sets and of
two 2-set8. Consider the two j>-sets
1 ) i^r — 1 ^4r -H 2 J i^r — 1 ^4r + 3 J ^4r + 2 ^4r + 3 )
1 J i^* — 1 ^4» + 2 ) i^« — 1 ^4» + 3 J k^ ^ 2 ^4» + 8 )
where 5 > r. All the terms of the first set are contained a« factors in each of
the terms p^^ik^^^y p,_i k^^^^^ which are of odd order in the k] consequently,
the multiplication is scalar. The term A:^, + 2 ^4* + 3 h^s no factor common with
the first set, and being of even order is commutative in regard to it. Hence
the two sets are commutative with one another. Next take the two j-sets
1 , Jr — 1 ^4r J Jr — 1 ^4r + 1 ) ^4r ^4r + 1 3
1 J 2« — 1 ^4» ) 2« — 1 ^4».+ H ^4» ^4» + 1 •
358 Clifford, Applications of Grassmann^s Extensive Algebra.
Here again all the terms of the first set are factors of j,_ i k^, and of q,__ik^^ij
and they have no factors in common with ^4,^44 + 1; since then all the terms
arc of even order in the k^ the multiplication is scalar. The proposition is
therefore proved.
We may set out a formal proof that the 2^ products of m terms, one
from each of the first m sets, are all distinct^ as follows : Suppose this true for
the first m — 1 sets : that is to say, that no two of the products formed from
them are either identical or such that their product is ± ^1 itj • • • ^-i*
Let then a , ft be two of these products ; and let Cj d be two terms of the next
set. Then we have to prove that ac can neither be equal to ± W, nor such
that the product acbd is ± k^ k2 . . . ^2m-i hm ^2m + i • Now if ao = db W,
multiply both sides by be; then ab = i cd. The product cd is one of the
terms of the new set ; it is either unity, or contains one or both of the new
units ^2inj ^2m + i> SO that it cannot be equal to ab. The product aftcrf cannot
he ± ^1 • • • ^2m + i unless cd is ^2m ^2« + i and ab is kik2 . . . ^2m-i j which is
contrary to the supposition. Hence if the products of the first m — 1 sets
are all distinct for the purpdses of the (2m — l)-way algebra, the products of
the first m sets will be all distinct for the purposes of the {2m + l)-way alge-
bra. But it is easy to see that the products of the first two sets are distinct.
Algebras with an even number of units.
Every algebra with 2m units is related to the adjacent algebra with 2m — 1
units in precisely the same way as biquaternions are related to quaternions ;
namely, it is simply that adjacent algebra multiplied by the double alge-
bra (1 , o) where o is the product of all the 2m units. For clearly all the
even terms of the (2m — l)-way algebra are also even terms of the
2m- way algebra, and so also are their products by o; but these are all
distinct from one another, and consequently are all the even terms of the
2m- way algebra.
The multiplication of o with the k of the (2m — l)-way algebra is scalar,
because the k are factors in the o, and they are both even terms.
Hence the 2m- way algebra is a product of the (2m— 1)- way algebra with
the double algebra (1 , o) , the two sets of units being commutative with one
another.
360 Craig, The Motion of a Point upon the Surface of an Ellipsoid.
Difierentiating the equation of the ellipsoid twice with respect to t, we have
X dx . y dy . z dz _ ^
d''di^ ¥dt'^ '^dt~ '
X cPx y tPy . z cPz
'^^It^'^T^W^ <? df
designate by P the last three terms of this last equation, then we have from (a)
(2) _P=ajp + /?.
Again, equations (1) give
\}-dx^ \^dy^d^,ldz^z\_ V^dx y^dy z^d^
WTt df'^ h^ dt df ^ <? dt di^A ~ '^ \-a^ dt "^ h* dt "^ &■ rfd
,o\xdxy^dyz_dz-\^
'^'^ La^dt^ ¥ dt^ <?dtl'
this is simply
Eliminating a from (2) and (3), we have
idp 1 dp_^
p dt ^ P + ^dt ~ '
the integral of which is
(4) p{P + ^)=A = const.
Multiplying equations {^)^y jr^ -^ j ^^^ -r: respectively, and adding, gives
czr ax az
dx cPx dy cPy dz cPz _ r. r dx dy dz'^
dt W "^ didt"^ '^dtdt'~^Vli'^'^'dt'^^'di\'
Integrating this we get
where B = const, of integration. For /? = or when no force acts towards
the center, we have the simple case,
s = s/B . t + const,
or the arc varies directly as the time. Any point on the ellipsoid can be
given as the intersection of this surface with the two confocals
—^ + _/_ + _ Jl_ - 1
dx zz —
Craig, The Motion of a Point upon the Surface of an Ellipsoid. 361
Then we have Xi and 7^ as the elliptic coordinates of the point, ;ii = const,
being the equation of one set of lines of curvature and T^.^ = const, the equa-
tion of the other set, the intersection of any two of which determines the
position of a point upon the surface. We have now (Salmon's Geom. of Three
Dimen., page 124),
x' + f + z' = a'+h' + c'-{\ + ^,) ;
also,
(6) di^—d(K'+du^+dz^=:hllh.\ ^^^' ^^^^ 1
and for the diflferentials dx^ dy^ dz the known values
and finally for the perpendicular from the center to the tangent plane
(8) ^4. it.4. ^ — hh — n
^^^ a'^ b*^ c*~ abc~P
From the expressions for dx, dy, and dz we can readily obtain
.Q d^ df rfz'_ A, — x^ r ^ d^ -\
^ ^ a» "^ i* "^ c* ~ 4 Ka'+;i,)(6'+;i,)(c^+;i,) (a^+ X2)(6*+Jl2)(cH ^2)J '
the first member of which equation, divided by dt^, is the quantity P, therefore
\dt) \dt)
rfz=--l '
(10) p _ .^1 - ^
_ * *
4
where for convenience we write
<D = (a* + X.) (ft* + ^i) (c' + ^.)
* = («« + \) {b' + :^){c' + ?^).
Substituting for P its value as obtained from equation (4), this last becomes
where D = abcA. Equation (5) becomes in elliptic coordinates
(12) ^ [| (§)■- 1 (1=)*] = <'-^ (^. + W .
where for brevity we write C = ^ («^ + ^^ + c^) + -B. Eliminate ^ from
91
362 Cbaig, The Motion of a Point upon the Surface of an Ellipsoid.
equations (11) and (12), and the result will be the differential equation, in
elliptic coordinates, of the path of the point upon the ellipsoid : this is
k,ir^) ^~^ — *">' ~ '■ ^^ ^' "^ '^^ U—- * J '
or
in which the variables are separated. If again we make /? = 0, that is, if
after the first impulse, no force acts upon the point but that in the direction
J)
of the normal, this equation becomes, on making — — zz 6*,
. _ y
V(a2 + ;ii)(6'' + ^i)l6-^ + J^i)l«'^ + J^i) V(a^ + ;i2)(6'' + ^)(^ + ;i2)(«'' + ^)
this is the differential equation of a geodesic upon the ellipsoid.
The integrals of these expressions will be elliptic for 6 = 0. If we change
the variables X^ and Xi into new ones defined by the equations
n •
our differential equation becomes
(14) '^
V(<i»j + i)(j«j + i)(c'j + \)(iye- <? + (3)
dri
^/{a\ + V)(p\ + 1) {i^n + 1) {Pn'-Cn-^^)
= 0.
C 3
Write 7) = ^ ^°^ T) ~ ^^ ^°^ call — ^1,-^2 J— >7i — >72 the roots of the
quadratic equations ^^ + ^^ + m = ,
rj^ + kri + mzzO'y
111
and further ~~ = a', -— 1= !/.—- = &. and our equation can be written
(t (r (T
(15)
/ ..3
V(^ + o-)!^ + ft')(^ + c)(^ + i,){i, + ^.)
= const.
/
V(>7 + a')(»7 + *')(»? + <^)(>7 + '7i)(»7 + »72)
These are not in general reducible to elliptic integrals. If now we find "K^ as
a function of ^i, say x^ =f(^i) ^/and * = F{^i) ^ F, then substituting in
364 CsAia, The Motion of a Point upon the Surface of an Ellipsoid.
And again,
r. \x dx ydy zdzl.
'^^'wTt^¥dt^&m:
or simply •
— P = op .
£!liminating a from these equations
1 dp 1 rfP _ 90
~p-dt^P~dt-'^^''
from which we have
^ + ^ = 2/?,<Z^ = 2 Crd^ + a ^ rfA .
'p if \ at ^
Now writing
2 JV(f5 = 2
we have as the integral of this equation
We can also readily obtain from the equations of motion the following
These equations give
dt
c^ \dt) ~ a •
which becomes in elliptic coordinates
Putting -y^j^ nz — ff^j and multiplying through by rf<^, W have after simple
reductions
s/<P {X, + e^) V* (Xj + 6') ~ '
or _ _
This is the diflferential equation of the path of the moving point which is,
as we see, a geodesic upon the ellipsoid.
366 Franklin, On a Problem of Isomerism.
not containing 1 by merging the I's into a single number. That we get no
repetitions is clear when we consider, first, that the same partition cannot
give two identical results, since in each result we have a diflferent number of
I's; and secondly, that if two identical results arose from two different par-
titions, we could, by merging the I's in each of these identical results, get
" back to the two diflferent partitions, which is absurd.
Now, the number of partitions of w — 1 is equal, as we have seen in A^
to the number of partitions of n which contain 1 ; that is, to the number of
partitions of n which contain one 1, plvs the number of partitions of n which
contain more than one 1 ; and we now see from B that this sum is the num-
ber obtained by the rule given at the outset.
The problem of isomerism above alluded to is as follows :
Required to find the number of diflferent compounds that can be formed
by wm-valent atoms and (m — 2) n + 2 univalent atoms; the word "com-
pound " being understood to mean any arrangement, whether conUnuous or not^
in which every atom appears, with exactly the number of bonds to which its
valence entitles it; it being understood, moreover, that no two univalent
atoms are connected with each other.
The problem reduces at once to the following : In how many ways can
n — 1 bonds be distributed among n atoms, no atom having more than m
bonds attached to it ? (Of course, it is understood that both ends of every
bond are attached.) A little consideration would show the truth of this ; but
it can very easily be formally proved, as follows :
Let X be the number of bonds connecting m-valent atoms with each
other ; then, since the whole number of bonds, counting those which connect
w-valent atoms twice^ is mw, the number of bonds which connect m-valent
atoms with univalent atoms — in other words, the number of univalent
atoms — is
mn — 2x •
But the number of univalent atoms is required to be {m — 2) n + 2. We
have, therefore, _
wn — 2ar n (m — 2) n + 2,
whence
X = n — 1.
368 FrankliiJ , Note on Indeterminate Exponential Forms.
If the union of the univalent atoms had been allowed, our result would
have been the number of partitions of n. For we would then have had to find
the number of ways in which n bonds can be distributed among n atoms,
and add this to the above. It is obvious that this number is the number
of partitions of n not containing 1 ; for if we had an isolated atom we could
not have n bonds, so that we can have no partitions containing 1; and for
any partition not containing 1, by making a closed circuit of atoms to cor-
respond to each number in it (and in no other way) we obtain n bonds, as
required. Now, from Ay the number of partitions of n not containing 1 is
the excess of the whole number of partitions of n over the whole number of
partitions of n — 1. Hence^ adding the number of partitions of n not con-
taining 1 to the whole number of partitions of n — 1, we obtain for our
result the whole number of partitions of n.
NOTE ON INDETERMINATE EXPONENTIAL FORMS.
By F. Franklin, Fellow of the Johns Hopkins University.
If y and z are two.functions of x^ which. become each equal to for a par-
ticular value of or, tf has, for that value, an indeterminate form ; and its value
is obtained through that of its logarithm. But we shall see that it is gene-
rally unnecessary to specially investigate the expression, its value being
always 1, provided that the ratio of z, or of some finite power of z, to y, is
not infinite.
For we know that the value of of when a: = is 1 ; hence we have
yf =z {tf'y = 1 , provided - is not infinite. If - is infinite, suppose — = a ,
where a is finite ; then we have •
unless k is infinite. Thaf^is, the value of the expression is always unity,
provided that the ratio of z^ or of some finite power of z^ to y, is not infinite.
This condition is, I believe, always fulfilled except in some cases when y or z
is itself an exponential or logarithmic function ; it is, at any rate, generally
easy to see at a glance whether or not such is the case.
Franklin, Note on Indeterminate Exponential Forms. 369
It follows from the above that the expression oo® is also always equal to
unity, with a similar restriction. For, if y" assumes this form for a particular
value of x^ put y zz — and we have y* =( — )=— ; and the value of this
u ^u/ u'
expression is, by the preceding, always 1, provided that the ratio of z^ or of
some finite power of z^ to u^ is not infinite ; or, in other words, provided that
the product of z^ or of some finite power of z, and y, is not infinite.
The examples given in most of the text-books I have seen, come within
the above restriction ; the following, for instance, are all the examples of the
above forms given by Williamson: ex. 1, p. 102: (sin x)^"" when xz=.0.
ex. 2, p. 103 : Tl + -) when or zz ; ex. 3, p. 103 : (-) when a: =z 0; ex;
^ x^ x^
30, p. 108: (l^^^i when x = oo; ex. 44, p. 109: (sin x^'' when or = 0.
In all these cases we at once recognize that the condition above found is fulfilled,
and the value of the function consequently 1.
In the following examples, found in Spitz's " DiflFerential- und Integral-
rechnung," the condition is not fulfilled, and the expressions require a special
L_ ^ / 1 \^
investigation: or^ + ^iogx ^^^n or = 0; ^j^"^^*^-^^ when or z= 0; f \^ when
a; = 00 .
It may be remarked, in this connection, that if f assumes, for a par-
ticular value of J?, the form 1* , its value can be very simply expressed
without taking its logarithm. Put zzz. — ; the value required is the limiting
_L _L ^ 1 Ay <^
value of ( 1 + Ay)^« = lim (1 + Ay)^^^" = lim [(1 + Ay)^^]^" = ^.
•s
A SYNOPTICAL TABLE OF THE IRREDUCIBLE INVARIANTS AND
CO VARIANTS TO A BINARY QUINTIC, WITH A SCHO-
LIUM ON A THEOREM IN CONDITIONAL
HYPERDETERMINANTS.
By J. J. Sylvester.
It is well known that every binary quintic can be expressed, and in only
one way, as the sum of three fifth powers of linear functions of its variables,
or which is the same thing, as the sum of the fifth powers of three yariables
connected by a linear equation, or finally, under the form
subject to the equation
^ + y + ^ = 0.
If 4) , 4' he any two covariants of a binary quintic in or, y, the most gene-
ral expression of the covariant produced by their operation on each other
through the variables is
/• d ' d\*
where i is any positive integer and and ^, y (abbreviations for ^ , y-J operate
on q> only whilst — , ^- operate on ^.
dx dy
Suppose now that 4), 4^ are expressed as functions, say 4>, *, of ^, y, 2,
between which there exists the linear relation Ix -{• my + nz =0 \ it may be
shown that the preceding expression becomes identical with
I
m
n
•
X
y
•
z
d
d
d
dx
dy
dz
4)*,
where or, y, z are to be treated as independent variables. In the present case,
therefore, writing •
r
A*4>*, or (which will be more convenient for v/riting) ?^A*4> will represent
the covariant derived from the alliance of 4> and * .
870
372 Sylvester, A Synoptical Table of the Irreducible Invariants^ dec.
(^ + y + ^) -^) the other special to those forms (such as 13 •!). which can be
obtained by the multiplication of lower forms (as 8*0, 5-1). Our object
must be to seek in all cases the simplest expressions that can be obtained.
2-2 = l-5A*l'5 ^i2 {abxy + acxz)^l,ab3iy .
I use the sign of equivalence to signify that numerical common multipliers
are to be rejected.
40 = 2-2A22-2
• • • • d d
= 2 (y — 5j) (z — x) {abxy + acxz + bcyz) — . — {abxy + acxz + bcyz)
= :Z { — ab + ac + be) ab = a^b^ + 6 V + c'a^ — 2abc {a + b + c)
1 • 5= ax* + 6/ + cz^
3*3= 2-2 A^ 1*5^2 «ar^(y — zy {abxy + bcyz -{' cazx) ^abcXx^.
Since or^ + / + z^= 3xyz + {x + y -{- z){x^ -^ ^ -\- z^ — xy — yz — zx) we have
{bis) 3*3.= abcxyz
5-1 = 3"3A*2*2^ abcX x{y — z)* {ahxy + hcyz + cazx) ^ ahcXbcx
2 • 6 = 1 • 6 A* 1 • 5 ^Sar* {y—'z)* {aif' + bf + cz») —Xaa^ {bi^-\-c^) = Xab a^f
3-5 = 2-2A l-5 = 2(afty + ac2)(y— z)(ax' + ft/ + c^»)
= 2 {ahf + acz) {by" — cz*) = taW — c'z^) + abet {zf —yz*)
4-4 = 3- 3AU-5 = a&c2ar {y—'zf {oaf + b^ + cz!") = abcX (bxy' + cx^)
= abcX [(<M^ + bi^ + cz*) X — <u?*] ^ abcXax*
5-3=2-2A 3-3 = 2(afty + <u?z)ty — 2)aic(a:» + y + z»)
^ aJc2 (y — «*) (afty + acz)*^ abcZax (ft/ — cz")
6-2 = 3-3A»3'3=a*J*c^ar {y — zy(3^ + f + :^) = <^¥(n{xy-^xz)
= aW<? {xy + yz ■\- zx) = €^¥6" {a^ -\- f + z"")
7- 1 = 4-4 A* 3-5 = a6c2 {{b + c) i*2 [(ajy — ac^z*) + aJc (z/ — yz*)}
= a*ft^c*2 (ft + c) x*2 (zy* — yz*) = a«ftV 2a (y — z)
8-0 =4- 4A*4-4 = a*ft^c*2a:r {y—'zf (<«:*+ fty* + cz*) = o^ft^c* (aft +«<?+ ftc)
4-6 = 3- 3A l-5 = aftc2a:'(y— z)(fl«r* + fty' + <?z»)=aft<r2a(y* — z*)a;*
6-4 = 2-2A4-4 = 2(a%+ acz) (y — z) aftc (<m?* + fty* + cz*)
= aftc (offty + acz){bf — c:^)
= abcl.{abY + oc^z*) + a'ft^c^ (zy» — yz») ,
which, since 2 {zf—y^) contains x-\-y-\ i^
= abet {c — ft) a*a?*
8-2 = 4-4a*4-6 = a*ft*c*2a (y — z)* {2a {f — z') x*} = (^Pc^ab {a^—f)
* For y* — f * I Bubetitute xx — xy.
Sylvester, A Synoptical Table of the Irreducible Invariants^ &c. 373
3-9=2-6Al-5 = 2 {abxY + «^^^) (y—^) («^ + */ + c^)
= 2 {abx'f + ac:x^z^) {b^ — cz')
= Xax" {by — &z') + abcx'fz'^ {zf — yz^) *
6-7 =4-4 A 1 • 5 = aftcSao:^ (y — 25) («x* + &/ + ^^) = «&^2a6 {x—y) o:^/
^7-5=4-4A 3-3 = a^6^c^aar'(^ — i)(j^ + / + z^)=a^6V2aj:'(/ — z')
11-1 =5-1 A 6-2 = a'^ftV26c(y — ;2)(a^ + / + z')=a'ftV2ftc(y — z)
9-3=6'2A3-3 = a'J«c^2j: (y — i)(r^+/+ 2') = a^b^(?{x—y){y—z){z—x)
12-0 = 6-2A'6-2 = a^ftV2(y — i) (a;' + y' + «') ^ (t*6V
13.1 = 7-lA6-2 = a^ftV2 (6 — c) (y — i) (4^ + 3^ + ^^^ _ a4j4^4(^_^) (y — ^)
= a^ftV (2 (6y + cz) — 2 (6z + cy))
= a^ftV (2 (ax + *y + ^^) + (ft^ + ^^ + dy)) = a^b^d'Xax
18-0 = 13-1 A 5 . 1 = a*ftV2a (y — i) (ftt?x + cay + abz) = a'¥c'l.a {c — b)
= a^iV {a — b) {b — c) {c — a) .
18*0 may also be obtained by the operation of 11*1 on 7*1, or instanta-
neously as the resultant of 1 • 5 , abcxyz and or + y + 5? . In the following
table the preceding results are collected ; for greater brevity instead of the
sign of summation I employ the sign + or — to signify respectively the sym-
metrical or semi-symmetrical completion of the terms to which it is aflSxed ;
m is used to signify abc.
1-2 abxy + : {a^b^ — 2ab(?) +
3-5 a^ + : m^ + , or inxyz : mbcx +
f abj^i^ + : a^ba^ + ^y^* — • ^^^^wr* + -
\ inabxy^ — : mV + - 'n^l>x — : vf^ab +
13-15 maa?y — : nu^cx^ — : m^abx^---
aVoc^y^ + rax^t/^z^ — : mabx^t^ — : m^ax^j^ — :
nv^bcy — : mVy — : m*
22 m^ax +
23 mV6 —
I propose, at some future time, to apply a similar method to obtain an
explicit representation of the irreducible forms appertinent to the binary
seventh ic, an arduous undertaking, but one that seems likely to lead to the
apperception of new forms of complex symmetry. The primitive may, for that
case, be represented by x'^ +y'^ -{-z^ + f^ connected by the linear equations
* Possibly this expression may be simplifiable by the addition of a suitable multiple of a? -|- y -f '•
94
16-21 I
374 Sylvester, A Synoptical Table of the Irreducible Invariants^ dc.
(if, m, w, p\x^ y^ ^j t) ==^0j (^, [ij v^ Ttlx, y, Zy ^) = , and A , the symbol of alii-
ance will be represented by
d d d d
dx dy dz dt
• • • •
X y z i
I m n p
X fl V 7t
Every in- and co-variant will then be a rational integer function of
Xj y, z^ t and the six minor determinants, which are the parameters of the line
represented by the above two linear equations.
It may be worth while to notice the representations of the irreducible
derivatives of the quartic when put under the indeterminate form ax^'\- by*-\'cz\
subject to the relation or tJ- y + ^ = . We get
2'0=l'4:A'l*4=Xa{y — zy{aa^+by^ + cz^)=ab + bc + ca
2 • 4=^1 • 4 A' 1 -4 = Soar" {y — zf {ax'' + by' + cz') = abx'y' + acx'z'' + %V
3 • = 1 • 4 A* 2 • 4 = 2a (3^ — i) ' {abx'y^ + acs'z^ + bcy'z'') = abc
3-6 = l-4A2-4=2ax2(^ — 2;)(2-4)
= 2 {a^bx^y — a^cx^z) + ahcxyzZ {yz^ — yh) .
As regards the sextic form, the first idea would be to regard it as the
resultant, in respect to one of the variables (say z)^ of the canonical system
discovered by me so long ago,
^ + ^jt + ^^^ + '^^y^ (^ — y)(y — ^) (^ — ^)
X +y + z,
but this will be found to give rise to expressions for the invariants and cova-
riants of extreme complexity. The representations will, I think, be simplified
by adopting the new canonical system .
^ + / + 2' + 37/i^yz1 (1)
ayz + bxz + cyx J (2)
and considering the sextic as the resultant of (1) and (2). It will then be
found that every covariant proper (calling its order, which is always an even
number, 2e) will still be a resultant of (2) and of some new form in x^ y, z of
order e.* The fact of the lowering, by one-half, the order of the form in
x^ y, z, corresponding to a covariant of any given order in a:, y, gives a
}
♦ For every quantic of an even order in ar, y is a ternary quantic in a?* + ajy , y* + yr, — ary , which quan-
tities are proportional to a;, y, 2 connected by the equation xy -{- xz ■\- yz =^ Q,
376 Sylvester, A Synoptical Table of the Irreducible Invariants^ Ac.
Let 4), *,..., © be the values of 4), 'v^, . . . , 6 expressed in terms of
Xj Pj . . . J t alone, and let
(ntnj . . . nj4>i*2 ... ©0
^Xjj ^y,» • . • J ^/,
be called D, it being understood that the meaning of any subscript, say jm, is
to cause the letters or, y, . . . , ^ to be changed into ^^ , y^? • . . , ^^. Again let
the operative determinant of the (i + j)th order written below
5,
(S.iV^). {S,N) {S.N).
be called J, (e being any of the suffixes 1, 2, 3, . . . , i) then it will be found
that to a numerical factor pres
(nr«n§-«. . . n?-')(j; + J2+ • • . + ^«)'(^i , 'J'i , . . . , e,) = 2>.
As a corollary, if the functions L, M, . . . , 2f^ are all linear in respect to
«,»,...,«, and if in respect to 1, «,«,..., 2 the resultants of ^, i, Jtf, . . . , iV;
'4' , £ , Jf , , . . , 2f; . . . are [<?)] , [*] , • . . , [0] (which is what we mean by
saying that ^,'^, , . . ,$ represent [<I>] , [*] , . . . , [0]) , it will be easily seen
to follow from the above theorem that the qth alliance of these quantics will
be itself represented hy
{Ji + J2 + + Jiy{'piAi,"',9,)* ^
* This expression may be put under the more compact form J*, J bein^ a matrix in which the first i lines
are the same as. those common to Jj , J, , . . . , «/<, and the lastj lines are the sums of the corresponding ones
in Jif J21 * ' • t Ji' Although I had submitted it to a mental process of demonstration (or what seemed such)
before sending it to the press, I am not without some little misgiving as to the exactitude of the theorem so far
as it regards the higher alliances ; for those of the first order it is easily verifiable, and, in that case, it should be
noticed that each of the i terms in the expression given by it will reproduce separately (but under quite h dis-
tinct form) the value of the Jacobian of ^, V>, . . . , Zr, . . . , iV. Some corresponding simplification in
practice, it is not improbable, will apply in the general case, supposing my doubts as to the validity of the
theorem to prove unfounded. It is important, and greatly enlarges the horizon of the subject, to remark that,
inasmuch as any ternary quadric is linearly tranfformable into the form xy'\-yz-\-zx^ it will follow that any
binary quanticof an even order, with its train of covariants, may be represented by corresponding ternary forms
of half their respective orders, combined with a perfectly general final conic, so that ex. gr. instead of the form
xy-\-yz-\'ZXj useful though it be as an intermediate step in the evolution of the theory, We may substitute the
handier and more advantageous one a;' -|-y' -f~ 2' ^^ ^^® auxiliary quadric.
378 Sylvester, A Synoptical Table of the Irreducible Invariants^ Ac.
cients of all orders, I do not know whether I shall be considered too bold or
fanciful in suggesting that there ought to exist, in the nature of things, some
theorem of development for several sets of variables analogous to Taylor's
for a single set : what such theorem is or could be I have at present no con-
ception, but as little, be it remembered, could any one, even Jacobi himself,
before the creation of hyperdeterminants, have had the remotest conception
in regard to a function of several variables bearing to ( — j <^ the same rela-
tion of analogy as the ordinary functional determinant to ,-^ , whether such
dx
function could exist, and if so, what it would be. I have always thought and
felt that beyond all others the algebraist, in his researches, needs to be guided
by the principle of faith, so well and philosophically defined as "the sub-
stance of things hoped for, the evidence of things not seen."
THE TANGENT TO THE PARABOLA.
By M. L. Holman and E. A. Engler, aS'^. Louis, Mo.
It is proposed in this paper to discuss, by the quaternion method, the
relations between three intersecting tangents to the parabola.
Suppose tangents to be drawn at any three points
as Ai^ Aii As] designate the vectors from the focus
to these points by pi, paj ps respectively. These tan-
gents intersect each other at Pi, P2iPz] designate
the vectors from the focus to these points of inter-
section by Tti, 7t2> ^3 respectively.
Let a be a unit vector parallel to the axis, and /?
a unit vector parallel to the directrix ; then the equa-
tion of the parabola referred to the focus as origin is
1^
4a
P = i:; y"^ +3^/^ -^^ = 4^ (/— 4^') « + y/^ '
(1)
in which p is any radius- vector and « = y SB =, the distance from the focus to
the vertex. The vector along the tangent is ^a + 3 .
2a
Since pi and p2 are vectors to the curve we have, by (1) ,
pi = ^ (y? — 4a') a +yi/? , P^ = j^ (3^ — ^a') a .+ !f2,3 ;
from which equations, since p^ = — (Tp)*,
Again
16a=
(2)
7t, = p2 + ^2Pi = ^(5^— 4a'') a+ffi^ + z (_yja + /?),
Tti = p3 + ^3 -Pi = 1^ (5^ — 4a*) a +ya/3 + X (— y^ + fi^,
in which x and z are unknown coefficients, easily determined by vector com-
parison, thus : x = ~ {yi—ys) , z = -i (yj —y^ , and
Tti =^ {y%yz—^^) a + J (^2 +5^3)/? ,
(3)
879
380 HoLMAN AND Engler, The Tangent to the Parabola.
whence, and by a cyclic change of suffices,
(T^)' = ^ (y§ + 4a^) (^ + 4a^) !► (4)
Hence, by (2) and the analogous formulae,
(T7tO^ = Tp.Tp3, (Tti,)^ = Tp3 Tp, , (TTta)' = Tp, Tp„ (6)
or the distance from the focus of a parabola to the intersection of two tangents is a
mean proportional between the radii-vectores to the points of contact.
Equation (3) shows that the distance of the point of intersection of two
tangents, from the axis, is the arithmetical mean between the ordinates to their
points of contact
From (5) we deduce
T7t2 TtTs = Tpx TtTx , T7t3 TtTi = Tp2 T7t2 , TtTi T^t^ = Tp3 TTta . (6)
By (3) and the analogous formulae
1 1
P2P3 = 7t3 — 7t2 = ^yi(y2— ys) « + g (^2—^3)/?,
^3 Pi = Tti — Pa z= — ^3 (ya — ya) « + ^ {Vi—y^ ^ •
Squaring and taking the tensors of these two expressions,
P2 Pa = ~ (ya-y2)^ (y? + 4a^), A, P, = -±-^ {y,-y,y (y§+ 4a^) . (7)
Then, by (4) and (7) ,
pTp: ^y^-y^y ' aTp: ~ (y^ -y^y ' ^ ^
and by (6) and (8) ,
P2 Pa TtTs T7t2 P3 Pi ITti ±7ts Pi P2 ±7t2 ATti ,^^
A^,~f^,~T^,' A^,~T^~T^,' 2;Ps~f;^~T^* ^^
From these equations the following relations between angles readily appear :
A.P^X* = PMs = A,SP, = P^SFi ,
SA,P^ z=SP2A^ = SPsP,,
SA.Pi z=SP2A, = SPiP,,
all of which, when Ai and A^ are fixed, are constant angles. Equations (9)
furnish a proof for a well-known method of constructing the parabola.
* The letter X, which through an oversight is wanting in the diagram, is any point of yl, P, extended.
• •
"" HoLMAN AND Engler, 7^ Tangent to the Parabola. 381
Thus far the consideration has been entirely general, the points of contact
being any points on the curve. The special cases are significant as indicative
of the neatness and simplicity of quaternion methods ; a few will therefore
be given.
1. When P2 is a multiple of pi, or when p2 — f)i is a focal chord, we have
Xf>2 n: pi , in which the sign of x is essentially negative.
Then by (1)
yi — ^a y2
which can only be satisfied, under the given conditions, by y^2 + 4a' = 0.
Then, by (1) and (3),
STtap, = - j^2 (y? + 4a')(5fiy2 + 4a'; = ,
or 7t3 is perpendicular to p^ ; ^. e. the line from the focus to the intersection of the
tangents at the extremities of a focal chord is perpendicular to the focal chord.
Also, by (5),
S (pi — Ttg) (p2 — Tta) = Spip2 + 7t^ = ,
or the tangents at the extremities of a focal chord are perpendicular to each other.
Since y^y^ + 4a' = , we have
7t3 = ^ (y,y2 — 4a') a + ~- (^1+^2) /? = — 2aa + 1 (^1+^2)/?,
«. e. the tangents at the extremities of a focal chord meet on the directrix.
2. When P2 becomes a multiple of ^,
p2 = — (y^ — 4a') a +^2/? = ^/?,
^ = y2 = ±2a;
i. e. the parameter equals the double ordinate through the focus, or twice the distance
from the focus to the directrix.
3. When p2 becomes a multiple 0/ a , ^2 = j ^^^
7t3 = — aa + l^i/?;
I. e. the subtangent is bisected at the vertex.
Also, 7t, — p, = — aa + ^(3-( f'~^'^ a+yi/^) =--^fia — ^y^,
S7t3 (7t3 — pi) 1= ;
I. e. flie perpendicular from the focus on a tangent m^ets it on the principal tangent.
96
382 HoLMAN AND Engler, The Tangent to the Parabola.
Again, the normal at A^ may be written
X7t^ = x{—aa + ^y^P)=za+y^^, x = 2\ z = — 2a\
showing that the subnormal is constant; and that the normal is twice the perpen-
dicular from the focus on the tangent
Also, xn^ = — za + p^, ^ (— ^a + ^^i/?) = — ^+|^ (y? — 4a2) a + y^^ ,
x = 2i z = ^{yi + 4a') = Tp,,
which shows that the distance from the foot of the normal to the focus equals the
radius vector to the point of contact, or the distance from the foot of the tangent to
the focuSy or the distance from the point of contact to the directrix; proofs for each
of which may also be obtained directly by writing the proper vector equalities
as obtained from the figure and determining the value of the unknown coeflS-
cients as above.
»
The part of the tangent from its foot to the point of contact is readily
found to be--^a+yi/?, since it is equal to ^^a + Pn and z has the value
— (^ + 4a^) J as already shown in our last equation. The part of the tangent
from the foot of the focal perpendicular to the point of contact is
_ 7t3 + p 1 = ^ 3^a + 1 y i/? ,
t. e, the tangent is bisected by the focal perpendicular. Whence the angle between
the tangent and the radius vector equals the angle between the tangent and the axisj
or the angle between the normal and the directrix ; and also, the tangent bisects the
angle between the diameter and the focal radius to the point of contact
The perpendicular from the focus on the normal is also — Tts + pi , whose
expression just given proves that the locus of the foot of the perpendicular
from the focus on the normal is a parabola , whose vertex is at the focus of the
given parabola, and whose parameter is-rof that of the given parabola.
The vector to the middle point of a focal chord (when pj is a multiple
of pi) is
P=-?-(p. + p.) = i(l + |) p.= 2^«-4a')p,
HoLMAN AND Engler, The Tangent to the Parabola, 383
that is, the locus of the middle points of focal chords is a parabola, whose vertex is
at the focus of the given parabola, and whose parameter is the parameter of the
given parabola.
4. When Tts becomes a multiple of (3;
^3 = j^ (^1^2 — 4a') a+ y (^1+^2)/? = ^^
••• yly2 = 4a^^ = -i(yl + y2) = 2^(3^ + 4a'),
When the point P3 coincides with the intersection of the tangent A^P^ with the
directrix, we have
7t3 = -2aa + ^^(y?-4a^)/3, 7^ = -^y, + 4ay, ^
hence, any tangent intersects the latus rectum and directrix in points equally dis-
tant from the focus.
m ^
ADDENDA TO MR. HALSTED'S PAPER ON THE BIBLIOGRAPHY
OF HYPER-SPACE AND NON-EUCLIDEAN GEOMETRY.
(Pages 261-276.)
Titles marked * were furnished by Mr. Halsted ; those marked f were giren in the Academy,
October 26th ; the others have been suggested by various individuals.
6.
*II (6). Note on Lobatchewsky's Imaginary Geometry. Phil. Mag.,
XXIX. 1865. pp. 231-233.
*II (e). On the rational transformation between two spaces. Lond. Math.
Soc. Proc. III. 1869-71. pp. 127-180.
9.
*II. On the degree of the surface reciprocal to a given one. [1855].
Irish Acad. Trans. XXIII. 1859. pp. 461-488.
10.
I. 2 vols., 4th and 5th Ed. Leipzig, 1874-5.
11.
* V. pp. 251-288. * VI. pp. 347-375. * VII. pp. XI-XIX ; also Nouv.
Ann. Math. IX. 1870. pp. 93-96, and Giornale di Mat. VIII. 1870.
pp. 84-89.
12.
*L pp. 185-204
♦III. pp. 232-255. For translations of II and III see 11, V and VI.
*IV. Teorema di geometria pseudbsferica. Giornale di Mat. X. 1872.
p. 53.
♦V. Sulla superficie di rotazione che serve di tipo alle superficie pseudo-
sferiche. Giornale di Mat. X. 1872. pp. 147-159.
13.
*I. Also in Nouv. Ann. Math. VII. 1868. pp. 209-221, 265-277; and
Napoli, Rendiconto VI. 1867. pp. 157-173.
3M
Addenda to Mr. ffalsted's Paper on the Bibliography of Hyper-Space, &c. 386
31.
*
I. Also Math. Ann , VII, pp. 531-537.
38.
Jena. 1873.
40.
*II. Ueber die Erzeugung der Curven dritter Classe und vierter Ord-
nung. Zeitschr. Math. Phys. XVIII. 1873. pp. 363-^86.
47.
Wien.
48.
fill. Address to the British Association at Dublin. August 14th, 1878.
55.
f First published in Proc. Lond. Math. Soc. Vol. VIII, pp. 57-59.
63. Agolini, G.
*I1 quinto postulate Euclidiano. Firenze. 1868.
64. BOUNIAKOFFSKY, V.
f Considerations sur quelques singularites . . . dans les constructions de
la geometrie non-euclidienne. Mem. de TAcad. de Peterb., serie VII, torn.
XVIIL 1872.
65. SCHMITZ-DUMONT.
fZeit und Raum in ihren denknothwendigen Bestimmungen abgeleitet
aus dem Satze des Widerspruchs. I. Leipzig. 1875.
66. MoNEO, 0. J.
fFlexure of Spaces. Lond. Math. Soc. Vol. IX.
67. Young, .G. P.
The relation which can be proved to subsist between the area of a plane
triangle and the sum of the angles on the hypothesis that Euclid's twelfth
axiom is false. Read before the Canadian Institute, 25th February, 1860.
{Further Addenda will he given in the next Volume.)
07
NOTES.
I.
Link' Work for x^.
{Extract from a Letter 0/ Professor Oayley to Mr. Sylvester.)
I suppose the following is substantially your link- work for r*. I use a
slot to make D move in the line OA ; but this could be replaced by proper
link-work. Supposing and A fixed; the line
OB is movable, and I wanted to have the distance
OB measured in a fixed direction. This can be
done by a hexagon OABQBA with equal sides,
and two other equal links BR , BR : then of course,
if 0, JS, Q are in linea, the hexagon will be sym-
metrical as to OQ^ and OB will be equal OB^ and
■^"^^ B' may be made to move in the fixed line OB.
BOA=\e, OA = AB = a, AC=CI) = ^ay
OB = 2acos\e, OI) = a{l + cos 0) = 2a cos* i ,
or " 2a.0D={pBy.
then
80th Noysmbkr, 1877.
II.
Link- Work for the Lemniscate.
{Extract from a Letter of Mr. A. W. Phillips of New Haven.)
In regard to the description of the Lemniscate I found in working upon
the 3-bar link- work, if and O were fixed points, ABOO a parallelogram,
^ AP= PB and OA=rz=i V2a, that P would describe the lemnis-
cate of which and O were the foci. The complete equation
^ "^ ' of the curve described by P expressed in bipolar coordinates
reduces to
(p^ + p'*-6a2)(pV^_aO = 0,
where p^ + p''^ — 6a^ = is the equation of a circle in bipolars. Then
py2 — ^4 -_ Q giy^s pp' — ^2 ^jj^ equation to the lemniscate.
8th Mat, 1878.
386
Notes. 387
III.
Euler^s Equations of Motion.
By James Loudon, University College] Toronto^ Canada.
1. A rigid body fixed at has at time t rotations, 6)1, 6)2, 6)3 round the
principal axes OA , OB ^ OC^ to determine the changes per unit time in these
rotations.
The positions 0A\ OB^ OC of the axes at time t-^-^t will be known from
the displacements in time S#, due to these rotations, of the points ^ (coi , 0, 0) ,
^(0, 6)2) 0) J C(0, 0, 0)3). The components of these displacements in the
directions OA , OB , OC, respectively, are evidently
, OiOsSt , — Oiojit , for A
— (dijidsStj 0, (d^idi^t^ for B
id^J^ty — (d^i^tj 0, for C.
The component rotations at time t + St are o^ + _! 5^, &c., which may be
represented by 0A\ OB, OC The changes of the rotations in time 8t are,
therefore, AA'j BBj CC Resolving these changes into the components
{AF, FP, PA') , {BG , GQ , QB) , (C//, HR , EC) in the directions of the
axes at time t, we get (observing that FP, PA' are the displacements in time
ht of the point i^(6)i + — ^ S^, 0, 0) , &c,, and neglecting infinitesimals of the
first order) the following as the resultant changes in time ^^:
AF+GQ + HB= (^i- - 6)^3 + "sWi) ^f=^j^t along OA
FP+BG+ RC - (w,w3 + ^ — 6)3(0,) Uz^^±U along OB
^ at y at
'PA' +QB + CH=(- 0,0, + COM + ^)^t = 4r ^t along OC'.
^ at / at
The changes per unit time are therefore ~ , ~ , ~ in the directions
at at at
OA , OjB, OC, respectively.
2. To determine the component changes of the body's moment of
momentum. At time t the components of the moment of momentum are
-4«j, Bid^, CL)3 in the directions of the principal axes, (where A, J9, C denote
the principal moments of inertia. At time t + ht the components are
-4 r^i + ~ Sn , &c., in the directions OJ', OB, OC Employing the figure
in a new sense, the former components may be represented by OA , OB , OC
and the latter by OA', OB., OC. The changes of the moment of momentum in
•time ht are therefore AA', BB, CC. Resolving these changes into their com-
ponents parallel to the axes at time t we get, as in the former case, (observ-
388
Notes.
ing that FP ^ PA are now the displacements in time ht of the point JP,
\Aidi + A -^ 3^j 0, 0) , &c.), the following as the resultant changes of the
moment of momentum in time ht:
[A — — Bojx^^ + Cidscj.j ht along OA
(AidiCd^ + B ~ — Cbgcoj) hi along OB
(
— Aidiid2 + Bid./di + ^-37) ^' along OC
The changes per unit time are therefore A -~ — {B — C) tOj^oj, &c., in the
dt
directions OA , OB ^ OC^ respectively.
21ST NOYKMBER, 1878.
IV.
Condition of a Straight Line Touching a Surface.
By James Loudon, University College, Toronto, Canada.
The following method of getting the condition of a straight line touching
a surface I have used for some years, and expected to see it in Salmon's 3d
edition. It was so short and unimportant that I never thought of publishing
it. (See Salmon's Geometry of Three Dimensions, Art. 80.)
Let u^a3i^'+bf+cz^+dw'+2fyz+2gzx+2hxy-\'2lxw+2myw+2nzwz=L(^^
then P^ax + ^y + yz + hw=zO, P^a'x + ^'y + yz + h'wzzO is a line touch-
ing u = 0. Therefore P — XP'= is, for some value of X, the tangent plane
at the point of contact {of, y", zf, vf) ; i. e.
(a — ;ia') x+{(3 — X/?') y+{y — ^y)z+{h — lh')w = and
{ax^ + h^ + gz:+lW)x+{hx' + bi/+fz:+mw')y+{gaf+fi/ + c^+nvf)z
+ (^r' + my" + wz' + dv/) wz=0 are identical. Therefore
aaf-\.ky' + gzf-\- lw' = k{a—7^') haf + by' +fzf+mv/ = k (^ —Xfi')
gx" +fy' + czf + nv/ = k ly — Xy) hf+my' + nzf + dvf =z k{h — kh')
ax^ + lSy + yzf + hv/zzO a'x'^ (3y + ysf + h'vf = .
Eliminating x", y", zf, v/, — k^ kX, we get the required condition
ay hy g J ly a , a'
h, b,f\ m,l3,iS'
9, /. <^. n, y, y
I, m, n, d , S, h'
a,/?,y, 5, 0,
a', /3', /, 5', ,
28th Noykmber, 1878.
= 0.
AM E R IC AN
Journal of Mathematics
PURE AND APPLIED.
Editor in Chief,
J. J. SYLVESTER, LL. D., F. R. S., Corr. Mem. Inst, of France.
Associate Editor ix Charge,
WILLIAM E. STORY, Ph. D., {Leipsic.)
W I t H the C o-o peratiox of
BEX JAMIX PEIRCE, LL. D., F. R. S., SIMON NEWCOMB, LL. D., F. R. S.,
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University, Supkrixtkndknt op the Amkrican Ephemkris,
In IVIechaiiics, In A-stronomya
and
H. A. ROWLAND, C. E.,
In Physics.
PlIBLlSHEU UNDER THE AUSPICP^S OF THE
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Note on a Class of Transformations which Surfaces may un<l(jrgo in S|mce of more
than Three Dimensions. By Simon Newcomb, 1
Researches in the Lunar Theory, I. Bv G. W. Hir.L, Nt/ack Turnpike , N. F., . 5
The Theorem of Three Moments. By Henry T. Eddy, UiuuerHity of Clacinnatiy . 27
Sohition of the Irreducible Case. By Guido Weichold, Ziitau, Saxoni/, . . 32
Desiderata and Suggestions. By PROPi-iisoR Cayley, Cambridge, England. No. 1. —
The Theory of Groui>s, 50
Note on the Theory of Electric AI>sorption, By H. A. Rowland, . *. .53
Esjwsizione del Metodo dei Minimi Quadrati. Per Annibale Ferrero, Tenente
Cnfonnello di Statu Maggiore^ ec. Firenze, IS70. By Cuarles S. Peiroe, New
York, 59
On an Ai)pli(^tion of the New Atomic Theory to the Graphical Representation of
the Invariants and Covariants of Binary Quantics. — With three Appendices and
an accompanying Plate (l^latu l\) By J. J. Sylvester, 64
Appendix L On Differentiants Expressed in Terms of the Differences of the
Roots t)f their Parent Quantics, .......... 83
Appendix :?. On M. Hermite's Ijaw of RcHjipnwity, . . . . .90
A M E R TC AlSr
Journal of Mathematics
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J. J. SYLVESTEK, LL. D., F. R. S., Gorr. Mem. fast, of France.
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{ ' (.) X T K X T S .
Oil an Application <»(' tlu» Ni?\v Atoniii' Tlu'ory to the Graphical Representation of the
Invariants and Covariants of IJinarv (iiianties. Bv J. J. Sylvkstek.
Appendix J. On I lermite's Law of Reciprocity, (Couchuleil,) .... 105
Note A, Completion of the Theory <»f Pi ineipal Forms, ..... 106
Note B. Additional Illustrations <»f the I-.a\v of Reciprocity, .... 107
Note C. On the I'rincipal Forms of the (ieneral Sextin variant to a C^uartic and
Quartin variant to a Sextie, . . . . . . . . . . .112
Note D, On the Probable Relation of the Skew Invariants of Binary Ciuinties and
Sexties to one another and to the Skew Invariant of the same Weight of the Binary
Nonic, . . . . . . . . . . . . .114
Appendix S, On Clebsi*h's '* Einfachstes System associirtcr Formen *' and its
Generalization, . . . . . . . . . .118
Note on the Ladenburg Carbon-graph, .125
Extract of a letter to Mr. Sylvester from Profe8.sor Clifford of University College,
London, . . . . . . . . . • .126
Researches in the Lunar Theory, II. By G. W. Hill, Nyack Turnpike, N. 1'., . . 129
Bipunctual Coordinates. (Plate II.) By F. Fu.vnklix, Fetfoic of the Johns Hopkim
Univerait};^ .............. 148
Desiderata and Suggestions. By Professor Caylky, (\iinbnd(je, Emjkind. No. 2. —
The Theory of Groups ; Graphical Representation, .174
On the Elastic Potential of Crystals. By William E. Story, 177
ThC'orie des Fonctions Num^riques Simplement P^ricMliques. (To be continued.) Par
Edouard Lucas, FrofeMeur au Lycte Charlemayne, Paris, . . . . .184
AMERICA^ISr
Journal of Mathematics
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Associate Editor in Charge,
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With the C o-o peration of
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to invito disaster. In any dilemma he needs the counsel of that class of his confreres whose labor has been in
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CONTENTS.
Paok
Th^orie des Fonctioiis Nuin^riques Simplement P^riodiques. (Suite.) Par Edouard
hvc\&, Profe88eur au Lycie' Oiarlemagiiey Paris, 197
The Elastic Arch. By Henry T. Eddy, ancinnaiiy O., 241
Researches in the Lunar Theory, II., (Continued.) By G. W. Hill, Hyack Timi-
pike, N. Y., 245
Bibliography of Hyper-Space and Non-Euclidean Geometry. By George Bruce
Halsted, Tutor in Princeton OoUeffe, N. J., 261
Some Remarks on a Passage in Professor Sylvester's Paper as to the Atomic Theory.
A Letter from Professor J. W. Mallet, of the University of Virginia, . . 277
' Notes :
I. Historical Data concerning the Discovery of the Law of Valence, . . . 282
II. On the Mechanical Description of the Cartesian. By J. Hammond, Baih,
Enaland, 283
III. A New Solution of Biquadratic Equations. By T. S. E. Dixon, Oiicago, Ills. 283
IV. On a Short Process for Solving the Irreducible Case of Cardan's Method. By
Otis H. Kendall, Aaaistant Professor in the University of Pennsylvania, . 285
V. An Extension of Taylor's Theorem. By J. C. G lash an, Ottawa, Canada, . 287
A.M ERIC A]Sr
Journal of Mathematics
PURE AND APPLIED.
Editor in Chief,
J. J. SYLVESTER, LL. D,, F. R. S., Gorr. Mem. Inst, of France.
Associate Editor in Charge,
WILLIAM E. story, Ph. D., (Leipsic.)
With the Co-operation of
BENJAMIN PEIRCE, LL. D., F. R. 8., SIMON NE WCOMB, LL. D., F. R. S.,
PR0FS880B OF MATHEMATICS IK HARVARD CORR. MeM. IN8T OF FRANCE.
U>'iYER8iTTy Superintendent of the American Ephemeeib,
In Meolianios, In A-stronomy,
AND
H. A. ROWLAND, C. E.,
In Physios.
Published under the Auspices of the
JOHNS HOPKINS UNIVERSITY.
ndvra ya [lav ra yiyvoityxdfieva dpi^fiov lj(pvri. — Philalaoa.
Vol Time I. Nnmber 4.
BALTIMORE:
Printed for the Editors by John Murphy & Co.
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